gauss-green theorem for weakly differentiable vector...

Gauss-Green Theorem for Weakly Differentiable Vector

Fields, Sets of Finite Perimeter, and Balance Laws

GUI-QIANG CHENNorthwestern University

MONICA TORRESPurdue University

AND

WILLIAM P. ZIEMERIndiana University

Abstract

We analyze a class of weakly differentiable vector fields F W RN ! R

N withthe property that F 2 L1 and div F is a (signed) Radon measure. These fieldsare called bounded divergence-measure fields. The primary focus of our inves-tigation is to introduce a suitable notion of the normal trace of any divergence-measure field F over the boundary of an arbitrary set of finite perimeter that en-sures the validity of the Gauss-Green theorem. To achieve this, we first establisha fundamental approximation theorem which states that, given a (signed) Radonmeasure � that is absolutely continuous with respect to HN�1 on R

N , any set offinite perimeter can be approximated by a family of sets with smooth boundaryessentially from the measure-theoretic interior of the set with respect to the mea-sure k�k, the total variation measure. We employ this approximation theoremto derive the normal trace of F on the boundary of any set of finite perimeterE as the limit of the normal traces of F on the boundaries of the approximatesets with smooth boundary so that the Gauss-Green theorem for F holds on E.With these results, we analyze the Cauchy flux that is bounded by a nonnegativeRadon measure over any oriented surface (i.e., an .N � 1/-dimensional surfacethat is a part of the boundary of a set of finite perimeter) and thereby developa general mathematical formulation of the physical principle of the balance lawthrough the Cauchy flux. Finally, we apply this framework to the derivation ofsystems of balance laws with measure-valued source terms from the formulationof the balance law. This framework also allows the recovery of Cauchy entropyflux through the Lax entropy inequality for entropy solutions of hyperbolic con-servation laws. c 2008 Wiley Periodicals, Inc.

Contents

1. Introduction 22. Radon Measures, Sets of Finite Perimeter,

and Divergence-Measure Fields 7

Communications on Pure and Applied Mathematics, Vol. 000, 0001–0063 (PREPRINT)c 2008 Wiley Periodicals, Inc.

2 G.-Q. CHEN, M. TORRES, AND W. P. ZIEMER

3. The Normal Trace and the Gauss-Green Formulafor DM1 Fields over Smoothly Bounded Sets 20

4. Almost One-Sided Smooth Approximationof Sets of Finite Perimeter 28

5. Main Theorem 336. The Divergence Measure of Jump Sets via the Normal Trace 407. Consistency of the Normal Trace with the Classical Trace 428. One-Sided Approximation of Sets of Finite Perimeter 459. Cauchy Fluxes and Divergence-Measure Fields 48

10. Mathematical Formulation of the Balance Lawand Derivation of Systems of Balance Laws 57

11. Entropy Solutions of Hyperbolic Conservation Laws 58Bibliography 60

1 Introduction

In this paper we analyze a class of weakly differentiable vector fields F W RN !

RN with the property that F 2 L1 and div F is a Radon measure � (that is, a

signed Borel measure with finite total variation on compact sets). These fields arecalled bounded divergence-measure fields, and the class is denoted by DM1. Theprimary focus of our investigation is to introduce a suitable notion of the normaltrace of any divergence-measure field over the boundary of an arbitrary set of finiteperimeter to obtain a general version of the Gauss-Green theorem.

Clearly, this investigation is closely related to the theory of BV functions in RN ;

in fact, it would be completely subsumed by the BV theory if the fields were of theform F D .F1; : : : ; FN ) with each Fk 2 BV.RN /, since div F D

PNkD1

@Fk

@xk

would then be a Radon measure (cf. [72, exercise 5.6]). However, in general, thecondition div F D � allows for cancellation, which thus makes the problem moredifficult and accordingly more important for applications (see Sections 9–11). Forthe Gauss-Green theorem in the BV setting, we refer to Burago and Maz0ja [10],Volpert [66], and the references therein. The Gauss-Green theorem for Lipschitzvector fields over sets of finite perimeter was first obtained by De Giorgi [24, 25]and Federer [30, 31]. Also see Evans and Gariepy [29], Lin and Wang [51], andSimon [64].

Some earlier efforts were made on generalizing the Gauss-Green theorem forsome special situations of divergence-measure fields, and relevant results can befound in Anzellotti [3] for an abstract formulation when F 2 L1 over a set withC 1 boundary and Ziemer [71] for a related problem for div F 2 L1; also see[1, 2, 5, 9, 27, 46, 47, 48, 50, 56, 57, 58]. In Chen and Frid [16, 19], an explicitway to formulate the suitable normal trace over a Lipschitz deformable surfacewas first observed for F 2 DM1. In particular, it has been proved in [16, 19] that

GENERALIZED GAUSS-GREEN THEOREM 3

the normal trace over a Lipschitz deformable surface, oriented by the unit normalvector �, is determined completely by the neighborhood information from the pos-itive side of the surface oriented by � and is independent of the information fromthe other side. This is the primary motivation for our further investigation intodivergence-measure fields. Chen and Torres [21] have established the normal tracefor any bounded divergence-measure field over a set of finite perimeter E and thecorresponding Gauss-Green theorem. One of the main results in this paper is to ob-tain this normal trace as the limit of the normal traces over the smooth boundariesthat approximate the reduced boundary @�E of E (see Definition 2.6). In partic-ular, the normal trace is determined completely by the neighborhood informationessentially from the measure-theoretic interior of the set (see Theorem 5.2), so thatthe Gauss-Green theorem holds for any set of finite perimeter.

We recall a very general approach, initiated by Fuglede [39], in which the fol-lowing result was established: If F 2 Lp.RN I R

N /, 1 � p < 1, is a vector fieldwith div F 2 Lp.RN /, then

(1.1)Z

E

div F dy D �

Z

@�E

F .y/ � �.y/dHN�1.y/

for “almost all” sets of finite perimeter E where HN�1 is the .N �1/-dimensionalHausdorff measure. The term “almost all” is expressed in terms of a conditionthat resembles “extremal length,” a concept used in complex analysis and potentialtheory (cf. [45, 68, 69, 70]). One way of summarizing our work in this paper is tosay that we wish to extend Fuglede’s result so that (1.1) holds for every set E offinite perimeter. Of course, this requires a suitable notion of the normal trace ofF on @�E. This is really the crux of the problem as F , being only measurable,cannot be redefined on an arbitrary set of dimensionN �1. The following exampleillustrates the subtlety of the problem.

Example. Let N D 2 and

F .x1; x2/ D

(

.1; 0/ if x1 > 0;

.�1; 0/ if x1 < 0:

Note that div F D 2H1 S with S D f0g � R, since, for any ' 2 C 10 .RN /,

hdiv F ; 'i D �

Z

RN

F � r' D

Z

fx1<0g

'x1�

Z

fx1>0g

'x1

D

Z

S

' dH1 C

Z

S

' dH1 D 2

Z

S

' dH1:


By letting EC D .0; 1/ � .0; 1/ and E� D .�1; 0/ � .0; 1/, there exist scalarfunctions f C and f �, respectively, on @�EC and @�E� such that

Z

E˙

' div F C

Z

E˙

F � r' D

Z

@�E˙

'f ˙dH1;

where

f C D

8

ˆ

<

ˆ

:

1 on f0g � .0; 1/;

0 on .0; 1/ � .0; 1/;

�1 on f1g � .0; 1/;

f � D

8

ˆ

<

ˆ

:

1 on f0g � .0; 1/;

0 on .�1; 0/ � .0; 1/;

�1 on f�1g � .0; 1/:

However, note that the normal traces f ˙ are not opposite on f0g � .0; 1/, as onemight at first expect.

To achieve the goal of this paper, we first establish a fundamental approximationtheorem which states that, given a Radon measure � on R

N such that � � HN�1,any set of finite perimeter can be approximated by a family of sets with smoothboundary essentially from the measure-theoretic interior of the set with respect tothe measure k�k (for example, we may take � D div F ). Then we employ thisapproximation theorem to derive the normal trace of F on the boundary of any setof finite perimeter as the limit of the normal trace of F on the smooth boundaries ofthe approximating sets and then establish the Gauss-Green theorem for F , whichholds for an arbitrary set of finite perimeter.

With these results on divergence-measure fields and sets of finite perimeter, weanalyze the Cauchy flux that is bounded by a nonnegative Radon measure � over anoriented surface (i.e., an .N �1/-dimensional surface that is a part of the boundaryof a set of finite perimeter) and develop a general mathematical formulation of thephysical principle of balance law through the Cauchy flux. In the classical settingof the physical principle of balance law, Cauchy [12, 13] first discovered that theflux density is necessarily a linear function of the interior normal (equivalently,the exterior normal) under the assumption that the flux density through a surfacedepends on the surface solely through the normal at that point. It was shown inNoll [54] that Cauchy’s assumption follows from the balance law. Ziemer [71]provided a first formulation of the balance law for the flux function F 2 L1

with div F 2 L1 at the level of generality with sets of finite perimeter. Alsosee Dafermos [22], Gurtin and Martins [42], and Gurtin and Williams [43]. One ofthe new features in our formulation is to allow the presence of exceptional surfaces,“shock waves,” across which the Cauchy flux has a jump. When the Radon measure� reduces to the N -dimensional Lebesgue measure LN , the formulation reducesto Ziemer’s formulation in [71], which shows its consistency with the classicalsetting. We first show that, for a Cauchy flux F bounded by a measure � , there


exists a bounded divergence-measure field F W RN ! R

N , defined LN -a.e., suchthat

F.S/ D �

Z

S

F .y/ � �.y/dHN�1.y/

for almost any oriented surface S oriented by the interior unit normal �. Thenwe employ our results on divergence-measure fields to recover the values of theCauchy flux on the exceptional surfaces directly via the vector field F . The valueas the normal trace of F on the exceptional surface is the unique limit of the normaltrace of F on the nonexceptional surfaces, which is defined essentially from thepositive side of the exceptional surface oriented by �. Finally, we apply this generalframework to the derivation of systems of balance laws with measure-valued sourceterms from the mathematical formulation of the physical principle of balance law.We also apply the framework to the recovery of Cauchy entropy flux through theLax entropy inequality for entropy solutions of hyperbolic conservation laws bycapturing entropy dissipation.

We observe the recent important work by Bourgain and Brezis [8] and De Pauwand Pfeffer [28] (see also Phuc and Torres [59]) for the following problem with adifferent point of view: Find a continuous vector field that solves the equation

(1.2) div F D � in �

for a given Radon measure �. In the case d� D f dx with f 2 LNloc.�/, theexistence of a solution F to (1.2) follows from the closed-range theorem as shownin [8]. It is proved in [28] that equation (1.2) has a continuous weak solution if andonly if � is a strong charge; i.e., given " > 0 and a compact set K � �, there is� > 0 such that

Z

�

� d� � "kr�kL1 C �k�kL1

for any smooth function � compactly supported on K.The organization of this paper is as follows. In Section 2, we first recall some

properties of Radon measures, sets of finite perimeter, and related BV functions,and then we introduce the notion of an oriented surface and develop some basicproperties of divergence-measure fields. In Section 3, we develop an alternativeway to obtain the Gauss-Green formula for a bounded divergence-measure fieldover any smooth boundary by a technique that motivates our further developmentfor the general case. In Section 4, we establish a fundamental approximation the-orem which states that, given a Radon measure � on R

N such that � � HN�1,any set of finite perimeter can be approximated by a sequence of sets with smoothboundary essentially from the interior of the set with respect to the measure k�k.In Section 5, we introduce the normal trace of a divergence-measure field F on theboundary @�E of any set of finite perimeter as the limit of the normal traces of F onthe smooth surfaces that approximate @�E essentially from the measure-theoretic


interior of E with respect to the measure kdiv F k, constructed in Section 4, andthen we establish the corresponding Gauss-Green theorem.

In Sections 6 and 7, we further analyze properties of divergence-measure fields,especially showing the representation of the divergence measures of jump sets viathe normal traces and the consistency of our normal traces with the classical traces(i.e., values) when the vector field is continuous. In Section 8, we first show that,if the set of finite perimeter E satisfies (8.1) (which is similar to Lewis’s uniformly

fat condition in potential theory [49]), there exists a one-sided approximation toE, and we then show that an open set of finite perimeter is an extension domainfor any bounded divergence-measure field. In Section 9, we first introduce a classof Cauchy fluxes that allow the presence of these exceptional surfaces or “shockwaves,” and we then prove that such a Cauchy flux induces a bounded divergence-measure (vector) field F so that the Cauchy flux over every oriented surface withfinite perimeter can be recovered through F via the normal trace over the orientedsurface.

In Section 10, we apply the results established in Sections 3 through 9 to themathematical formulation of the physical principle of balance law and the rigorousderivation of systems of balance laws with measure-valued source terms from thatformulation. Finally, in Section 11, we apply our results to the recovery of Cauchyentropy flux through the Lax entropy inequality for entropy solutions of hyperbolicconservation laws by capturing entropy dissipation.

Added in proof. We recently learned of the paper by Šilhavý [63], which hassome overlap with our work. However, the techniques involved are completely dif-ferent, thus offering the interested reader more depth and insight into the problem.Theorem 3.2 in [63] gives conditions under which the measure � will vanish onsets of appropriate Hausdorff dimension. This result is established in Lemma 2.25of the present paper, where the optimal condition is expressed in terms of capac-ity. Section 4 in [63] begins with several results of the divergence theorem in itsgeneral form as a linear functional on the space of Lipschitz functions on @� asin [3], and then Šilhavý proceeds to obtain the result, theorem 4.4, which is thecounterpart to theorem 2 in Chen and Torres [21] and is related to Theorem 5.2 ofthis paper, which contains several results of descending generality that also yieldtheorem 2 in [21]. The methods of the present paper are substantially different anduse exclusively the methods of geometric measure theory to obtain Theorem 5.2.

Furthermore, our normal trace of a divergence-measure field on the boundaryof a set of finite perimeter is derived as the limit of the normal traces on the smoothboundaries of approximate sets. These approximate sets essentially belong to themeasure-theoretical interior of the set of finite perimeter with respect to the di-vergence measure. Such a geometric interpretation has important applications inconservation laws. The present paper also contains results that are of interest inthemselves, such as a completely self-contained proof of the divergence theoremin the case of a C 1 boundary (Theorem 3.3), the fact that the normal trace of a


divergence-measure field is the classic dot product when the field is continuous(see Theorem 7.2), and the fundamental theorem for almost one-sided approxima-tion of sets of finite perimeter (Theorem 4.10). Even though the second part oftheorem 5.2 of [21] is essentially the same as Theorem 9.4 of this paper, our resultsin Sections 9 and 10 highlight the importance of the normal traces of divergence-measure fields, in conservation laws, defined on every set of finite perimeter (versusalmost every set). Also, see [61, 62] for related papers.

2 Radon Measures, Sets of Finite Perimeter,

and Divergence-Measure Fields

In this section we first recall some properties of Radon measures, sets of finiteperimeter, and related BV functions (cf. [2, 29, 33, 40, 72]). We then introducethe notion of oriented surfaces and develop some basic properties of divergence-measure fields. For the sake of completeness, we start with some basic notions anddefinitions.

First, denote by HM theM -dimensional Hausdorff measure in RN forM � N ,

and by LN the Lebesgue measure in RN (recall that LN D HN ). For any LN -

measurable set E � RN , we denote jEj as the LN -Lebesgue measure of the set

E and @E as its topological boundary. Also, we denote B.x; r/ as the open ball ofradius r and center at x. The symmetric difference of sets is denoted by

A�B WD .A n B/ [ .B n A/:

Let � � RN be open. We denote by E b � that the closure of E is compact

and contained in �. Let Cc.�/ be the space of compactly supported continuousfunctions on � with k'k0I� WD supfj'.y/j W y 2 �g:

DEFINITION 2.1 A Radon measure on � is a signed regular Borel measure whosetotal variation on each compact set K b � is finite, i.e., k�k.K/ < 1. Thespace of Radon measures supported on an open set � is denoted by M.�/. If� 2 M.�/ does not take negative values, then we refer to such � as a nonnegativeRadon measure.

Any Radon measure � can be decomposed into the difference of two nonnega-tive Radon measures � D �C � ��; the total variation of � is k�k D �C C ��.Equivalently, if� is a Radon measure on�, the total variation of� on any boundedopen set W � � is equal to

k�k.W / D sup

� Z

�

' d� W ' 2 Cc.W /; k'k0I� � 1

�

D sup

� 1X

iD0

j�.Bi /j

�

;

(2.1)

where the second supremum is taken over all pairwise disjoint Borel sets Bi withW D

S1iD1Bi . Since the space of Radon measures can be identified with the dual


of Cc.�/, we may consider a Radon measure � as a linear functional on Cc.�/,written as

(2.2) �.'/ WD

Z

�

' d� for each ' 2 Cc.�/:

We recall the familiar weak-star topology on M.�/, which, when restricted to asequence f�kg, yields

�k�* � in M.�/I

that is, �k converges to � in the weak-star topology if and only if

(2.3) �k.'/ ! �.'/ for each ' 2 Cc.�/:

The spaceLp.�;�/, 1 � p � 1, denotes all the functions f with the propertythat jf jp is �-integrable. The conjugate of p is p0 WD p=.p � 1/. The Lp-normof f on a set E with integration taken with respect to a measure � is denoted bykf kpIE;�. In the event � is Lebesgue measure, we will simply write kf kpIE .

THEOREM 2.2 (Uniform Boundedness Principle) Let X be a Banach space. If Tkis a sequence of linear functionals on X that converges weak-star to T , then

lim supk!1

kTkk < 1:

Next, we quote a familiar result concerning weak-star convergence.

LEMMA 2.3 Let � be a Radon measure on �, and let �k be a sequence of Radon

measures converging weak-star to �. Then we have the following:

(i) If A � � is any open set and �k are nonnegative Radon measures,

�.A/ � lim infk!1

�k.A/:

(ii) If K � � is any compact set and �k are nonnegative Radon measures,

�.K/ � lim supk!1

�k.K/:

(iii) If k�kk�* � , then k�k � � . In addition, if the �-measurable set E b �

satisfies �.@E/ D 0, then

�.E/ D limk!1

�k.E/:

More generally, if f is a bounded Borel function with compact support in

� such that the set of its discontinuity points is �-negligible, then

limk!1

Z

�

f d�k D

Z

�

f d�:


DEFINITION 2.4 For every ˛ 2 Œ0; 1� and every LN -measurable set E � RN ,

define

(2.4) E˛ WD fy 2 RN W D.E; y/ D ˛g;

where

(2.5) D.E; y/ WD limr!0

jE \ B.y; r/j

jB.y; r/j:

Then E˛ is the set of all points where E has density ˛. We define the measure-

theoretic boundary of E, @mE, as

(2.6) @mE WD RN n .E0 [E1/:

DEFINITION 2.5 A function f W � ! R is called a function of bounded variation

if each partial derivative of f is a Radon measure with finite total variation in �.Notationally, we write f 2 BV.�/: Let E b � be an LN -measurable subset. Wesay that E is a set of finite perimeter if �E 2 BV.�/. Consequently, if E is aset of finite perimeter, then r�E is a (vector-valued) Radon measure whose totalvariation, denoted by kr�Ek, is finite.

DEFINITION 2.6 Let E b � be a set of finite perimeter. The reduced boundary of

E, denoted as @�E, is the set of all points y 2 � such that

(i) kr�Ek.B.y; r// > 0 for all r > 0, and(ii) the limit

�E .y/ WD limr!0

r�E .B.y; r//

kr�Ek.B.y; r//

exists and j�E .y/j D 1.

The set @�E is also called the perimeter of E and its .N � 1/-Hausdorff mea-sure is denoted by P.E/ WD HN�1.@�E/. The following result was proved byDe Giorgi [24] (see also [2, theorem 3.59] and [72, theorem 5.7.3]).

THEOREM 2.7 LetE be a set of finite perimeter. The reduced boundary ofE, @�E,

is an .N � 1/-rectifiable set, which means that there exists a countable family of

C 1 manifolds Mk of dimension N � 1 and a set N of HN�1-measure zero such

that

(2.7) @�E �

� 1[

kD1

Mk

�

[ N :

Moreover, the generalized gradient of �E enjoys the following basic relationship

with HN�1:

(2.8) kr�Ek D HN�1 @�E


and, for HN�1-a.e. y 2 @�E,

limr!0

kr�Ek.B.y; r//

˛.N � 1/rN�1D 1;

where ˛.N � 1/ is the Lebesgue measure of the unit ball in RN�1.

The unit vector �E .y/ is called the measure-theoretic interior unit normal to

E at y (we sometimes write � instead of �E for notational simplicity). In view ofthe following, we see that � is aptly named because � is the interior unit normal toE provided that E (in the limit and in measure) lies in the appropriate half-spacedetermined by the hyperplane orthogonal to �; that is, � is the interior unit normalto E at x provided that

D�

fy W .y � x/ � � > 0; y … Eg [ fy W .y � x/ � � < 0; y 2 Eg; y�

D 0:

The following result is due to Federer (see also [72, lemma 5.9.5] and [2, theo-rem 3.61]).

THEOREM 2.8 If E b � is a set of finite perimeter, then

(2.9) @�E � E1=2 � @mE; HN�1.� n .E0 [ @�E [E1// D 0:

In particular, E has density either 0 or 12

or 1 at HN�1-a.e. x 2 �, and HN�1-

a.e. x 2 @mE belongs to @�E.

Remark 2.9. In view of Definition 2.5, (2.8), and (2.9), it is clear that, if E b �

is a set of finite perimeter, then HN�1.@mE/ < 1: Conversely, it was proved byFederer (see [33, theorem 4.5.11]) that, if HN�1.@mE/ < 1, then E is a set offinite perimeter.

We will refer to the sets E0 and E1 as the measure-theoretic exterior and inte-

rior of E. We note that, in general, the sets E0 and E1 do not coincide with thetopological exterior and interior of the set E. The sets E0 and E1 also motivatethe definition of measure-theoretic boundary. We note that (2.9) implies that, forany set E b � of finite perimeter,

� D E1 [ @�E [E0 [ N

where HN�1.N / D 0. If we define a set E to be “open” if E is both measurableand D.E; x/ D 1 for all x 2 E, then this concept of openness defines a topology,called the density topology. It is an interesting exercise to prove that the arbitraryunion of open sets is also open; the crux of the problem is to prove that the arbi-trary union is, in fact, measurable. This topology is significant because it is thesmallest topology (the one with the smallest number of open sets) for which theapproximately continuous functions are continuous [41].

The following result, which is easily verified (although tedious), will be neededin what follows.


LEMMA 2.10 If A;B b � are sets of finite perimeter, then

@m.A \ B/ � .@mA \ B/ [ .A \ @mB/ [ .@mA \ @mB/:

DEFINITION 2.11 Let � 2 C1c .RN / be a standard symmetric mollifying ker-

nel; that is, � is a nonnegative function with support in the unit ball and satisfiesk�k1IRN D 1. With u 2 L1.RN /, we set u" WD u � �", where the sequence�".y/ WD 1

"N �.y"/ forms a mollifier.

Recall the following properties of mollifications (cf. [72]).

LEMMA 2.12

(i) If u 2 L1loc.RN /, then, for every " > 0, u" 2 C1.RN / and D˛.�" � u/ D

.D˛�"/ � u for each multi-index ˛.

(ii) With k WD 1"k

and "k ! 0, uk.x/ ! u.x/ whenever x is a Lebesgue point

of u. In particular, if u is continuous, then u" converges uniformly to u on

compact subsets of RN .

Remark 2.13. Functions in the spaces BV.RN / and W 1;p.RN /, 1 � p � 1,have precise representatives; that is, if u 2 BV.RN /, then there is a function u� 2

BV.RN / such that u and u� are equal a.e. and that the mollification sequence of u,uk , converges to u� at all points except those that belong to an exceptional set Ewith HN�1.E/ D 0. However, this is not the same as saying that u has a Lebesguepoint, which is slightly stronger.

A similar statement is true for functions in the Sobolev space W 1;p.RN /, 1 <p � 1, except that the exceptional set E has p-capacity zero; see Definition 2.21below. As we will see, the 1-capacity vanishes precisely on sets of HN�1-measure zero. Thus, we can say that functions in the spaces BV and W 1;p haveprecise representatives that are defined, respectively, 1 and p almost everywhere.

DEFINITION 2.14 If u = �E for a set of finite perimeter E, we denote uE as thecorresponding precise representative u�.

Indeed, when u is taken as �E , Lemma 2.12 can be considerably strengthened:

LEMMA 2.15 If uk is the mollification of �E for a set of finite perimeter E, then

the following hold:

(i) uk 2 C1.RN /.

(ii) There is a set N with HN�1.N / D 0 such that, for all y … N , uk.y/ !

uE .y/ as k ! 1 and

uE .y/ D

8

ˆ

<

ˆ

:

1; y 2 E1,12; y 2 @�E,

0; y 2 E0:

(iii) ruk�* ruE in M.RN /.


(iv) If U is an open set with kruEk.@U / D 0, then krukk.U / ! kruEk.U /

as k ! 1.

(v) r�E D ruE .

PROOF: Only (iii) requires a proof, since (i), (ii), and (iv) are results fromstandard BV theory and (v) is immediate from the definitions and the fact thatuE D �E almost everywhere. As for (iii), since uk ! uE in L1.RN /, thenuk ! uE when considered as distributions, which implies that ruk ! ruE asdistributions and consequently as measures since ruk;ruE 2 M.RN /. �

The next result affirms the notion that the mollification is generally a norm-reducing operation.

LEMMA 2.16 Let E be a set of finite perimeter, and let uk denote the mollification

of �E . Then

krukk1 � kr�Ek:

PROOF: For any f 2 BV.RN /, consider the convolutions

f".y/ WD

Z

RN

�".y � x/f .x/dx:

Using rf" D �" � .rf / and f" 2 C1.RN /, we obtain

rf".y/ D

Z

RN

�".y � x/dm.x/

where m WD rf is the measure. Thus, we have

jrf".y/j �

Z

RN

�".y � x/ dkmk.x/:

In particular, when f D �E and f"kD uk with "k D 1

k, then m D r�E and

jruk.y/j �

Z

RN

�"k.y � x/dkmk.x/ for all y 2 R

N :

That is,Z

RN

jruk.y/jdy �

Z

RN

Z

RN

�"k.y � x/dkmk.x/dy

D

Z

RN

Z

RN

�"k.y � x/dy dkmk.x/ � kmk.RN /:

�


We recall that the BV space in fact represents equivalence classes of functionsso that, when a function in a class is changed on a set of LN -measure zero, itremains in this class. The same is true for sets of finite perimeter because, bydefinition, the characteristic function �E of a set of finite perimeter E is a functionof bounded variation. Thus, it follows that E may be altered by a set of LN -measure zero and still determine the same essential boundary @mE. Throughout,we will choose a preferred representative for E and thereby adopt the followingconvention.

DEFINITION 2.17 E WD fy W D.E; y/ D 1g [ @mE.

DEFINITION 2.18 A vector field F 2 Lp.�I RN /, 1 � p � 1, is called a

divergence-measure field, written as F 2 DMp.�/, if � WD div F is a Radonmeasure with finite total variation on � in the sense of distributions. Thus, for' 2 C1

c .�/, we have

�.'/ WD .div F /.'/ D �

Z

�

F � r' dy:

The total variation of � is a nonnegative measure which, for any open set W � �,is defined as

k�k.W / WD supf�.'/ W k'k0I� � 1; ' 2 C1c .W /g

D sup

�Z

�

F � r' dy W k'k0I� � 1; ' 2 C1c .W /

�

:

A vector field F 2 DMploc.�/ means that, for any W b �, F 2 DMp.W /.

DEFINITION 2.19 Let F 2 DMp.�/, 1 � p � 1. For an arbitrary measurableset E b �, the trace of the normal component of F on @E is a functional definedby

(2.10) .TF /@E .'/ D

Z

E

r' � F dy C

Z

E

' d�

for any test function ' 2 C1c .�/. Clearly, .TF /@E is a distribution defined on�.

Note that this definition assumes only that the set E is measurable. Later, we willprovide an alternative definition when E is a set of finite perimeter (see Theo-rem 5.3).

PROPOSITION 2.20 Let E b � be an open set. Then spt..TF /@E / � @E. That

is, if and ' are test functions in D.�/ with D ' on @E, then .TF /@E . / D

.TF /@E .'/.

PROOF: If the support were not contained in @E, there would be a point x0 …

@E with x0 2 spt..TF /@E /\E. This implies that, for each open set U containingx0, there exists a test function ' 2 C1

c .U\E/ such that .TF /@E .'/ ¤ 0. Choose


U so that U � RN n @E. Let F " denote the mollification of F (see Lemma 2.12).

Then, since spt.F "'/ b E,

0 D

Z

E

div.F "'/ dy D .TF "/@E .'/

D

Z

E

F " � r' C

Z

E

' div F " dy

!

Z

E

F � r' dy C

Z

E

' d� D .TF /@E .'/ ¤ 0;

where we have used @E \ spt.'/ D ¿ in the limit. Thus, we arrive at our desiredcontradiction. �

DEFINITION 2.21 For 1 � p � N , the p-capacity of an arbitrary set A b RN is

defined as

(2.11) p.A/ WD inf

�Z

�

jr'jp dy

�

;

where the infimum is taken over all test functions ' 2 C1c .�/ that are identically 1

in a neighborhood of A. It is well known (cf. [34]) that p.A/ D 0 for 1 < p < Nimplies that HN�pC".A/ D 0 for each " > 0 and that, conversely, if HN�p.A/ <

1, then p.A/ D 0. In view of Remark 2.13 and Lemma 2.25, it is easy to verifythat the class of competing functions in (2.11) can be enlarged to the Sobolev spaceW 1;p.�/.

Remark 2.22. The case of p D 1 requires special consideration. In 1957, Flem-ing conjectured that 1.A/ D 0 if and only if HN�1.A/ D 0. This was settledin the affirmative by Gustin [44], who proved the boxing inequality, from whichFleming’s conjecture easily follows (cf. [35]).

The next results are basic (cf. [69, 70, 72]).

PROPOSITION 2.23 Let p be the p-capacity defined in (2.11). Then

(i) If Ek � RN is a sequence of arbitrary sets, then

p.lim infk!1

Ek/ � lim infk!1

p.Ek/:

(ii) If E1 � E2 � � � � are arbitrary sets, then

p

� 1[

kD1

Ek

�

D limk!1

p.Ek/:

(iii) If K1 � K2 � � � � are compact sets, then

p

� 1\

kD1

Kk

�

D limk!1

p.Kk/:


(iv) If fEkg is a sequence of Borel sets, then

p

� 1[

kD1

Ek

�

� limk!1

p.Ek/:

(v) If A � RN is a Suslin set, then

supf p.K/ W Kcompact � Ag D inff p.U / W U open � Ag:

Any set function satisfying conditions (i)–(iv) is called a true capacity in the sense

of Choquet and a set A satisfying condition (v) is said to be -capacitable.

Remark 2.24. One of the main reasons for studying the capacity is its importantrole in the development of Sobolev theory. It was first shown in [34] that everyfunction u 2 W 1;p.�/ has a Lebesgue point p-a.e. In particular, in view ofRemark 2.22, this implies that a function u 2 W 1;1.�/ has a Lebesgue pointeverywhere except for an exceptional set E with HN�1.E/ D 0. In case u 2

BV.�/, we have a slightly weaker statement than the corresponding one for u 2

W 1;p.�/:

limr!0

«

B.x; r/u.y/dy D u�.x/ for HN�1-a.e. x 2 �.

It turns out that the Sobolev space is the perfect functional completion of thespace C1

c .�/ relative to the p-capacity. See [4] where the concept of perfectfunctional completion was initiated and developed.

LEMMA 2.25 If F 2 DMploc.�/,

NN�1 � p � 1, then kdiv F k � p0 I that is, if

B � � is a Borel set satisfying p0.B/ D 0, p0 WD p=.p� 1/, then kdiv F k.B/ D

0. In particular, when p D 1 (i.e., p0 D 1/, Remark 2.22 implies that kdiv F k �

HN�1.

PROOF: Because of the inner regularity of kdiv F k and condition (v) of Propo-sition 2.23, it suffices to show that �.K/ D 0 for any compact set K � B , where� WD div F . Since p0.K/ D 0, then there exists a sequence of test functions'k 2 C1

c .�/ (see, for example, lemma 2.2 in [6]) such that

(i) 0 � 'k � 1 and 'k D 1 on K,(ii) kr'kkp0I� ! 0 as k ! 1,

(iii) 'k.y/ ! 0 as k ! 1 for all y 2 � except those in some set A � � with p0.A/ D 0, and

(iv) 'k is supported in an open setOk withO1 � O2 � � � � � K andT

k Ok D

K.

Since div F D �, we have

�.K/C

Z

�nK

'k d� D �

Z

�

F � r'k dy � kF kpI� kr'kkp0I�:


Thus, using (i), (ii), and (iv), we conclude

j�.K/j � k�k.Ok nK/C kF kpI�kr'kkp0I� ! 0 as k ! 1:

�

COROLLARY 2.26 If F 2 DMploc.�/,

NN�1 < p < 1, and if HN�p0

.B/ < 1

for B � �, then p0.B/ D 0 and hence kdiv F k.B/ D 0.

Remark 2.27. If F 2 DMp.�/, NN�1 < p � 1, with div F D �, then, in

view of the fact that ' 2 W 1;p0.�/ is defined HN�p0

-a.e. and therefore �-a.e., itfollows that the integral

Z

�

' d�

is defined and is meaningful.

DEFINITION 2.28 A compact set K � RN is called a p-removable set for F 2

Lp.RN /, 1 � p � 1, provided that F 2 DMp.RN /whenever F 2 DMp.RN n

K/.

THEOREM 2.29 If K � RN is compact with HN�p0

.K/ D 0, 1 � p0 < N , then

K is a p-removable set for F 2 DMp.

PROOF: It suffices to show that, for each test function 2 C1c .RN /,

(2.12) �. / D �

Z

RN

F � r dy:

Let 'k be the sequence defined in the proof of Lemma 2.25 that satisfies (i)–(iv).We approximate by a sequence

(2.13) k WD .1 � 'k/ 2 C1c .RN nK/:

Since F 2 DMp.RN nK/, we have

(2.14) �. k/ D �

Z

RN

F � r k dy:

From the fact that 'k ! 0, p0-a.e. (and therefore �-a.e., by Lemma 2.25 andCorollary 2.26), we obtain that k ! , �-a.e., and therefore Lebesgue’s domi-nated convergence theorem yields

�. k/ WD

Z

RN

k d� !

Z

RN

d� WD �. /:


On the other hand, the Hölder inequality and (ii)–(iii) in the proof of Lemma 2.25imply

Z

RN

F � r k dy !

Z

RN

F � r dy;

and thus, we conclude with our desired result (2.12).�

EXAMPLE 2.30 (Chen and Frid [16]) Denote U the open unit square in R2 that has

one of its sides contained in the line segment

L WD fy D .y1; y2/ W y1 D y2g \ @U:

Define a field F W R2 n L ! R

2 by

F .y/ D F .y1; y2/ D

�

sin

�

1

y1 � y2

�

;� sin

�

1

y1 � y2

��

:

Clearly, F 2 L1.R2/, and a simple calculation reveals that div F D 0 inRN n L. Then F belongs to DM1.R2), but the field is singular on one side L

of @U , and therefore F is undefined on @U ; it has no trace on @U in the classicalsense. Note also that the points of L are all essential singularities of F because thefollowing limit does not exist:

limy!x

F .y/ for y 2 R2 n L; x 2 L;

and therefore the normal trace of F on @U is given by

limt!0

Z

@Ut

F .y/ � �.y/dH1.y/ D limt!0

Z

Ut

div F dy D 0;

where Ut WD fy 2 U W dist.y; @U / > tg. Thus, we have shown the following:

(1) F is an element of DM1.R2/, while each component function of F isnot in BV.R2/;

(2) F has an essential singularity at each point of L and therefore cannot bedefined on L;

(3) as we will see later, Theorem 5.3, F has a weak normal trace on L that issufficient for the Gauss-Green theorem to hold.

For more properties of the spaces DMp of divergence-measure vector fields, seeChen and Frid [16, 19].

The following theorem provides a product rule for the case p D 1. For thesake of completeness, we will include its proof, which is slightly different fromthat given in [16]. We denote by fgkg the sequence of C1

c .RN / mollificationswith the property that gk ! g in L1.RN / and such that krgkk ! krgk (cf. [72],p. 500).


THEOREM 2.31 (Chen and Frid [16]) Let F 2 DM1.RN / and g 2 BV.RN / be

bounded with compact support. Then

(2.15) div.gF / D g� div F C F � rg;

where F � rg denotes the weak-star limit of the measures F � rgk , and g� is the

precise representative of g.

PROOF: Let F " be the mollification of F and set � WD div F . Since F " aresmooth, the classical product rule yields

(2.16) div.gkF "/ D gk div F " C F " � rgk :

First, we note that div F " D .div F /" D �"�* � in M.RN / as " ! 0. Since

gkF " ! gkF in L1loc.RN / as " ! 0, we obtain from (2.16) that, in the sense of

distributions,

(2.17) F � rgk D div.gkF / � �.gk/:

Owing to the fact that F 2 L1, we see that F � rgk is a bounded sequence inL1.RN / and hence there is a subsequence such that F � rgk converges weak-starto some measure, denoted by F � rg. Letting k ! 1 in (2.17) yields

(2.18) F � rg D div.gF / � �.g�/: �

The next result, Federer’s co-area and area formulas (see [33, cor. 3.2.20]), willbe of critical importance to us in what follows.

THEOREM 2.32 (Federer [32]) Suppose that Y and X are Riemannian manifolds

of dimension N and k, respectively, with N � k. If f W Y ! X is a Lipschitz

map, then

(2.19)Z

Y

g.y/Jf .y/dHN .y/ D

Z

X

� Z

f �1.x/

g.y/dHN�k.y/

�

dHk.x/

whenever g W Y ! R is HN -integrable. Here, Jf .y/ denotes the k-dimensional

Jacobian of f at y, namely, the norm of the differential df .y/ of f at y. Alter-

natively, it is the square root of the sum of the squares of the determinants of the

k � k minors of the differential of f at y.

Furthermore, if k � N , the following area formula holds:

(2.20)Z

Y

Jf .y/dHN .y/ D

Z

X

N.x/dHk.x/ D

Z

f .Y /

N.x/dHk.x/;

where N.x/ denotes the (possibly infinite) number of points in f �1.x/.

In the event that u 2 BV.RN /, there is another version of the co-area formuladue to Fleming and Rishel [36].


THEOREM 2.33 If u 2 BV.RN /, then

kruk.RN / D

Z 1

�1

P.fu > tg/dt:

Conversely, if u 2 L1.RN / withR 1

�1 P.fu > tg/dt < 1, then u 2 BV.RN /.

LEMMA 2.34 Let u W RN ! R be a Lipschitz function, and let A � R

N be a set

of measure zero. Then

HN�1.u�1.s/ \ A/ D 0 for almost all s:

PROOF: This is immediate from the co-area formula:

0 D

Z

A

jru.y/jdy D

Z

R

HN�1.A \ u�1.s//ds:

�

One of the fundamental results of geometric measure theory is that any set offinite perimeter possesses a measure-theoretic interior unit normal � that is suitablygeneral to ensure the validity of the Gauss-Green theorem.

THEOREM 2.35 (De Giorgi and Federer [24, 25, 30, 31]) If E has finite perimeter,

thenZ

E

div F dy D �

Z

@�E

F .y/ � �.y/dHN�1.y/

whenever F W RN ! R

N is Lipschitz.

The De Giorgi–Federer result shows that integration by parts holds on a verylarge and rich family of sets, but only for fields F that are Lipschitz. As we ex-plained in the introduction, the Gauss-Green formula for BV vector fields oversets of finite perimeter was treated in [10, 66]. We contrast their results with thefollowing result by Fuglede [39].

THEOREM 2.36 (Fuglede [39]) Let F 2 Lp.RN I RN /, 1 � p < 1, with div F 2

Lp.RN /. Then there exists a function g W RN ! R with g 2 Lp such that

(2.21)Z

E

div F D �

Z

@�E

F .y/ � �.y/dHN�1.y/

for all sets of finite perimeter E except possibly those for whichZ

@�E

g.y/dHN�1.y/ D 1:

The following, which is a direct consequence of Fuglede’s result, will be of useto us. Suppose that u W R

N ! R is Lipschitz. For s < t , consider the “annulus”AsIt WD fx W s < u.s/ � tg determined by u. Then, by appealing to the co-area


formula, we see that AsIt is a set of finite perimeter for almost all s < t . Moreover,again appealing to the co-area formula, we see that, for almost all s < t ,

(2.22)Z

AsIt

div F D �

Z

@�AsIt

F .y/ � �.y/dHN�1.y/:

One of the main objectives of this paper is to demonstrate that, even when F 2

DM1.RN /, we can extend Fuglede’s result by showing that (2.21) and (2.22)hold for all sets of finite perimeter, not merely for “almost all” sets in the senseof Fuglede [39]. Although we don’t employ Fuglede’s theorem directly, his resultprovided the motivation and insight for the development of our method.

The Gauss-Green formula for bounded divergence-measure fields over sets offinite perimeter was obtained in Chen and Torres [21]. The product rule fromLemma 2.31 was used to prove that

(2.23) .div F /.E1/ WD

Z

E1

div F D �

Z

@�E

2�EF � ruE ;

where 2�EF � ruE is the weak-star limit of the measures 2�EF � ruk . Anotherobjective of this paper is to obtain the trace measure as the limit of the normal traceover smooth boundaries that approximate @�E.

3 The Normal Trace and the Gauss-Green Formula

for DM1 Fields over Smoothly Bounded Sets

In this section we obtain the normal trace and the corresponding Gauss-Greenformula for a bounded divergence-measure field over any smoothly bounded set.Although Lemma 3.1 below is sufficient to obtain our main result, the reason whywe give a proof of Theorem 3.3, which is much stronger than what our developmentrequires, is twofold: First, it gives a self-contained treatment of the Gauss-Greenformula for C 1 domains; second, it reveals the general outline of our method usedto obtain the main result, Theorem 5.2. It also underscores the fact that the normaltrace is indeed an interior normal trace in the sense that our definition is determinedby the behavior of F in the interior ofU (see Definition 3.4). This method consists,roughly speaking, in approximating the boundary of the given set by a family ofsuitable surfaces for which the Gauss-Green theorem holds and then obtaining thedesired trace as the weak limit of the normal trace over the approximating surfaces.

LEMMA 3.1 Let F 2 DM1loc.R

N / whose distributional divergence is a measure

�, and let F " be a mollification of F . Then the classical divergence theorem holds

whenever E b RN is a set of finite perimeter, namely,

(3.1)Z

E

div F " D �

Z

@�E

F ".y/ � �.y/dHN�1.y/:


If, in addition, we assume the two conditions

(i) F " ! F HN�1-a.e. on @�E and

(ii) �.@E/ D 0,

then

(3.2) �.E/ D �

Z

@�E

F .y/ � �.y/dHN�1.y/:

PROOF: First, the classical Gauss-Green formula (3.1) holds because F " is

smooth (in particular, Lipschitz). Furthermore, since �" WD div F "�* div F D �

in M.RN / and since F " @�E is uniformly bounded, we obtain from (3.1) that

�".E/ WD

Z

E

div F " D �

Z

@�E

F ".y/ � �.y/dHN�1.y/

! �

Z

@�E

F .y/ � �.y/dHN�1.y/;

�".E/ ! �.E/ (by assumption (ii)):

This establishes our result. �

The importance of this result is that, with assumptions (i) and (ii), we obtainthe Gauss-Green theorem for all sets of finite perimeter whenever F is a bounded,measurable vector field with div F D �. As stated earlier, our main objective is toobtain the same result without assuming (i) and (ii) by defining a suitable notion ofnormal trace for F on @�E.

DEFINITION 3.2 Given a compact C 1 manifold M , we define the exterior deter-mined byM to be that (connected) component U of R

N nM that is unbounded. Theinterior determined by M , U , is defined to be everything else in the complementof M ; namely,

U D

1[

kD1

Bk with Bk � RN nM is a bounded component.

Thus,

RN nM D U [

� 1[

kD1

Bk

�

D U [ U:

THEOREM 3.3 Let U � RN be the interior determined by a compact C 1 manifold

M of dimension N � 1 with HN�1.M/ < 1. Then, for any F 2 DM1loc.R

N /,


there exist a signed measure � supported on @U D M with � � HN�1 @U and

a function Fi � � W @U ! R such that, for any ' 2 C 1c .RN /,

Z

U

div.'F / D

Z

U

' div F C

Z

U

F � r'

D �

Z

@U

' d� D �

Z

@U

'.Fi � �/.y/dHN�1.y/;

(3.3)

and

kFi � �k1 � CkF k1;

where C is a constant depending only on N and U .

DEFINITION 3.4 With �.y/ denoting the interior unit normal to M at y, we mayregard Fi � � as the interior normal trace of F on @U and thus write

.Fi � �/.y/ D F .y/ � �.y/:

Hence, with this convention, it is convenient to abuse the notation and write (3.3)as

(3.4)Z

U

div.'F / D �

Z

@U

'F .y/ � �.y/dHN�1.y/;

while bearing in mind that, since F is merely a measurable field and thus definedonly up to a Lebesgue null set, it may not even be defined on @U . We use the term“interior normal trace” to suggest that Fi � � is determined by the behavior of F inthe interior determined by the manifold M . The proof will reveal that, in a similarway, it is possible to define the concept of “exterior normal trace.” This will bediscussed more fully below in Theorem 5.2.

First, we adapt a result of Whitney [67] to our situation in which the open set Uis the interior determined by a compact C 1 manifold M of dimension N � 1 withHN�1.M/ < 1. Whitney’s result states that an .N � 1/-dimensional manifold ofclass C 1 has a C 1 vector field that is almost normal.

THEOREM 3.5 [67, theorem 10A, p. 121] Suppose that M WD @U is an .N � 1/-

dimensional compact C 1 manifold in RN and let ˛ > 0. Then there exist a unit

C 1 vector field, ƒ� W M ! RN , and a number 0 < ı < 1 (that depends on ˛/

with the following properties:

(i) If ƒ�.p/ D v, then j�p.v/j � ˛jvj, where �p W RN ! Tp.M/ denotes

the orthogonal projection onto the tangent plane of M at p. Thus, ƒ�.p/

is close to �.p/ when ˛ > 0 is small, and

S�p WD fq 2 R

N W q D tƒ�.p/; 0 < t < ıg � U:

We think of the vectors Q�p WD ıƒ�.p/ as quasi-normals and observe that

jQ�p j D ı:


(ii) As p ranges over M , the segments S�p fill out a neighborhood U �

ıof M in

a one-to-one way. That is,

U �ı D

[

p2M

S�p ;

where S�p1

\ S�p2

D ¿ whenever p1; p2 2 M with p1 ¤ p2.

(iii) The mapping �� W U �ı

! M defined by

��.q/ WD p if q 2 S�p

is of class C 1 and has the property that

�.q/ WD j��.q/ � qj � 2 dist.q;M/ for q 2 U �ı .

PROOF OF THEOREM 3.3: The proof is divided into ten parts.

Step 1. We begin with some preliminaries:(1) The mapping �� W U �

ı! @U D M may be considered as the projection of

U �ı

onto @U along the quasi-normalQ�p , and the number �.q/ is the distance

from q to M , measured along S�p where ��.q/ D p.

(2) The open set Ut is defined, for all 0 < t < ı, as

Ut WD U n fq 2 U �ı W �.q/ < tg:

All of the open sets Ut , 0 < t < ı, are nested and contained in U withlimt!0 Ut " U .

(3) Using the fact that � is continuous, we have

@Ut � . �/�1.t/ for all t ;

with equality holding whenever t is not a critical value of �.(4) We define Mt WD @Ut . It will be shown that Mt is a C 1 manifold in Step 3.

Step 2. The open sets Ut are sets of finite perimeter for almost all t 2 .0; ı/.From (iii) in Theorem 3.5, we know that �� is of class C 1 on U �

ı; thus, so is

� .U �ı

n M/. Since � is C 1 on the open set U �ı

, it is therefore only locallyLipschitz. We may employ the co-area formula in Theorem 2.33 to conclude that,for any compact set A of finite perimeter, with u WD � A 2 BV.RN /,

Z 1

0

HN�1.@Ut \ A/dt D

Z 1

0

HN�1.u�1.t/ \ A/dt

D

Z

A

jrujdx � Lip.u/LN .U �ı / < 1:

Step 3. For all t 2 .0; ı/, @Ut WD Mt is a manifold of class C 1 and there exists

C.@U;N / independent of t such that

(3.5) HN�1.@Ut / D H

N�1.. �/�1.t// � C.@U;N /HN�1.@U /:


The set Mt WD @Ut may be considered as a deformation of M D @U along thevector field Q�. To see this, consider the C 1 mapping ht W @U D M ! @Ut D

Mt defined for 0 < t < ı as

q WD ht .p/ D tƒ�.p/ 2 @Ut ;

so that h0.p/ D p and ��.q/ D p with jht .p/ � pj D t .For each 0 < t < ı, the map ht W M ! Mt is clearly of class C 1; the

Jacobian Jht depends only on t and kDƒ�k1. Therefore, since h is univalent, wemay invoke (2.20) to conclude the following global bound whenever A � M is anHN�1-measurable set:

(3.6)

HN�1.ht .A// D H

N�1.ht .M \ A//

D

Z

A

Jht dHN�1.x/

� C.t; kDƒ�k1/HN�1.A \M/

� C.ı; kDƒ�k1/HN�1.A/:

Since �� ı ht D I (i.e., the identity), the chain rule implies that the differential ofht is nonsingular everywhere on M . Thus, we see that ht is a diffeomorphism andhence that Mt is an .N � 1/-manifold of class C 1. Then (3.5) is immediate from(3.6).

Observe also that there exists a set N � .0; ı/ of L1-measure zero that includesthose countable values of t for which kdiv F k.. �/�1.t// ¤ 0. Also, referring toLemma 2.34, it is clear that the values of t for which condition (i) in Lemma 3.1fails will be included in N and, for the rest of the proof, we consider only valuesof t outside of the exceptional set N.

Step 4. The signed measures defined by

(3.7)

�t .B/ WD

Z

B\@Ut

F .y/ � �.y/dHN�1.y/ for each Borel set B � RN ;

D

Z

B\u�1.t/

F .y/ � �.y/dHN�1.y/ for a.e. t (by Theorem 2.32);

along with �Ct and ��

t , where �t D �Ct � ��

t , all converge, for a suitable subse-

quence tk ! 0, weak-star to

(3.8) .�Ctk; ��tk; �tk /

�* .�C; ��; �/ in M.RN /:

Since F 2 L1 and HN�1.@U / < 1, we see from (3.5) that HN�1.@Ut / � C

for some C > 0 independent of t , which yields that the measures �t , t > 0, form


a bounded set in M.RN /. Hence there exist a sequence ftkg ! 0 and Radonmeasures � , �C, and �� with � D �C � �� such that

(3.9) .�Ctk; ��tk; �tk /

�* .�C; ��; �/ in M.RN /:

It should be kept in mind that the convergence in (3.9) means, for example, that

(3.10) �tk .'/ ! �.'/ for each test function ' 2 Cc.RN /:

Step 5. The supports of the measures �C, ��, and � are all contained in @U .

To prove the claim for �C, we assume the contrary, and let x 2 spt.�C/ n @U

and choose B.x; r/ such that B.x; r/ \ @U D ¿. Since x 2 spt.�C/, there exists' 2 C.B.x; r// such that

R

' d�C WD �C.'/ ¤ 0. Then, since ' is continuous,we find that �C

tk.'/ ! �C.'/ ¤ 0. This implies that �C

tk.'/ ¤ 0 for all small

tk > 0, which leads to a contradiction since Utk � U , spt.�Ctk/ � @Utk , and

@Utk \B.x; r/ D ¿. The proof of the claim for �� and � can be done in the sameway.

Step 6. With � representing any of the three measures �C, ��, and � , we have

(3.11) limtk!0

�tk .@Utk / ! �.M/ D �.@U /:

First, let � WD �C. Observe that spt �tk � @Utk implies

(3.12) limtk!0

�tk .@Utk / ! �.@U /:

By Lemma 2.3 and Step 5,

(3.13) lim inftk!0

�tk .@Utk / D lim infk!1

�tk .RN / � �.RN / D �.@U /:

Now choose a compact setK � @U [@Utk . Then, again by Lemma 2.3 and Step 5,we obtain

lim suptk!0

�tk .@Utk / D lim suptk!0

�tk .K/

� �.K/ D �.@U / since K contains @U :(3.14)

Thus, we obtain our desired result, (3.11), by (3.13) and (3.14). By taking � tobe the other positive measure � D ��, we obtain (3.11) for ��, and consequently(3.11) holds for � as well.

Step 7. The measure � is well-defined.


This can be seen as follows: Let Ut 0k

be another sequence of open sets to whichLemma 3.1 applies. Then, assuming that tk > t

0k

for all k, we have

�.Ut 0k

n Utk / D

Z

Ut0k

div F �

Z

Utk

div F

D �

Z

@Ut0k

F .y/ � �.y/dHN�1.y/C

Z

@Utk

F .y/ � �.y/dHN�1.y/

D ��t 0k.@Ut 0

k/C �tk .@Utk /:

Since Ut 0k

n Utk � U n Utk is a monotone decreasing sequence of sets with U n

Utk ! ¿, it follows that k�k.Ut 0k

n Utk / ! 0 and therefore that �t 0k.@Ut 0

k/ �

�tk .@Utk / ! 0, which shows that � is well-defined.

Step 8. � � HN�1 @U .LetA � @U with HN�1.A/ D 0. From (2.8), we have that kr�U k D HN�1

@U and hence kr�U k.A/ D 0. From general measure theory, we have

(3.15) 0 D kr�Ek.A/ D inffkr�Ek.G/ W A � G; G openg

and thus there exists an open set G � RN , G � A, such that HN�1.G\ @U / < ".

Moreover, using (3.6), we obtain

k�tk k.G/ �

Z

G\@Utk

jF .y/ � �.y/jdHN�1.y/

� kF k1HN�1.G \ @Utk /

� C.N; @U /kF k1HN�1.h�1

tk.G \ @Utk //;

where limtk!0HN�1.h�1

tk.G \ @Utk // D HN�1.G \ @U /. Then,

k�k.A/ � k�k.G/ � lim inftk!0

k�tk k.G/ D CkF k1HN�1.G [ @U /

� "CkF k1:

Since " is arbitrary, we conclude k�k.A/ D 0, as desired.

Step 9. We apply Lemma 3.1 to the divergence-measure field 'F and take lim-

its to obtain (3.3).From Theorem 2.31, it follows that, if ' 2 C 1c .R

N /, then 'F is also a diver-gence-measure field and

(3.16) div.'F / D ' div F C F � r':


An application of Lemma 3.1 shows that

(3.17)Z

Utk

div.'F / D �

Z

@Utk

'F .y/ � �.y/dHN�1.y/ for all tk … N;

where �' WD div.'F / is a Radon measure. Therefore, since the sets Utk areincreasing, we have

(3.18) limtk!0

�'.Utk / ! �'.U /:

Thus, using (3.10) and (3.18) and letting k ! 1 in (3.17) yields

(3.19)Z

U

div.'F / D �

Z

@U

' d�:

Step 10. The Radon-Nikodym derivative of � with respect to HN�1 @U ,

T .y/, is by definition the normal trace of F , denoted by Fi � �.y/, and enjoys the

bound

jFi � �.y/j � C.@U;N /:

Since � � HN�1 @U , the Radon-Nikodym theorem implies that there exists anHN�1-integrable function T W @U ! R such that (3.19) can be written as

Z

U

div.'F / D �

Z

@U

'T .y/dHN�1.y/:

Note that T is the Radon-Nikodym derivative of � with respect to HN�1 @U

whose value at HN�1-a.e. y 2 @U can be determined by Besicovitch’s differenti-ation theorem [7]:

jT .y/j D limr!0

ˇ

ˇ

ˇ

ˇ

�.B.y; r//

HN�1.@U \ B.y; r//

ˇ

ˇ

ˇ

ˇ

:

Since the balls B.y; rj / can be chosen such that k�[email protected]; rj // D 0, we have

jT .y/j D limrj !0

limtk!0

ˇ

ˇ

ˇ

ˇ

�tk .B.y; rj //

HN�1.@U \ B.y; rj //

ˇ

ˇ

ˇ

ˇ

D limrj !0

limtk!0

ˇ

ˇ

ˇ

ˇ

R

@Utk\B.y;rj /

F .y/ � �.y/dHN�1.y/

HN�1.@U \ B.y; rj //

ˇ

ˇ

ˇ

ˇ

� kF k1 limrj !0

limtk!0

HN�1.@Utk \ B.y; rj //

HN�1.@U \ B.y; rj //

� C.@U;N /kF k1 limrj !0

HN�1.@U \ B.y; rj //

HN�1.@U \ B.y; rj //(by (3.6))

D C.@U;N /kF k1:

�


Remark 3.6. In particular, when U is of class C 2, the interior normals to @Uthemselves do not intersect in a sufficiently small neighborhood of @U . Therefore,in the above proof, one may directly use the interior normals to @U as ƒ�

p .

4 Almost One-Sided Smooth Approximation

of Sets of Finite Perimeter

We now proceed to establish a fundamental approximation theorem for a setof finite perimeter by a family of sets with smooth boundary essentially from themeasure-theoretic interior of the set with respect to any Radon measure that isabsolutely continuous with respect to HN�1. That is, we prove that, for any Radonmeasure � on R

N such that� � HN�1, the superlevel sets of the mollifications ofthe characteristic functions of sets of finite perimeter provide an approximation bysmooth sets that are k�k-almost contained in the measure-theoretic interior of E.This approximation and Lemma 3.1, after passage to a limit, lead to our mainresult, Theorem 5.2.

LEMMA 4.1 Let � be a Radon measure on RN such that � � HN�1. Let E

be a set of finite perimeter, and let uk be the mollification of �E . Then, for any

t 2 .0; 1/ and AkIt WD fy W uk.y/ > tg, there exist " D ".t/ and k� D k�."; t/

such that

(i) k�k.AkIt nE/ < " if t 2 .0; 12/ and k � k�I

(ii) k�k.AkIt nE1/ < " if t 2 .12; 1/ and k � k�I

(iii) k�k.E1 n AkIt / < " if t 2 .12; 1/ and k � k�I

(iv) k�k.E n AkIt / < " if t 2 .0; 12/ and k � k�.

PROOF: We first show (ii). With t 2 .12; 1/, choose 0 < " < t � 1

2. Since uk

is the mollification of �E , we know that uk.y/ ! uE .y/ for HN�1-a.e. y andtherefore the same is true for k�k as well. By Egorov’s theorem, for any " > 0,there exist k� D k�."; t/ and an open set U" such that k�k.U"/ < " and thatjuk.y/ � uE .y/j < " for all y 62 U" and for all k � k�. On AkIt n U", we have

t < uk.y/:

Since uk.y/ < uE .y/C " on RN n U", we have

1

2< t � " < uE .y/ H) uE .y/ D 1 H) y 2 E1:

This yields

AkIt n U" � E1 H) AkIt nE1 � U":

Since k�k.U"/ < ", our desired result (ii) follows.For the proof of (i), given t 2 .0; 1

2/, we choose 0 < " < t and proceed as

above.


We next show (iv). With t 2 .0; 12/, choose 0 < " < 1

2� t . For all large k, we

have juk.y/ � uE .y/j < " for all y 62 U". Thus, on E n U",

1

2� uk.y/ � uE .y/ � uk.y/ < "

H) uk.y/ >1

2� " > t for all y 2 E n U";

which impliesEnU" � AkIt I therefore,EnAkIt � U" and thus k�k.EnAkIt / < ".

For the proof of (iii), given t 2 .12; 1/, we choose 0 < " < 1� t and proceed as

above. �

COROLLARY 4.2 For each t 2 .0; 12/ and " < minft; 1

2� tg, there exists k� D

k�."; t/ > 0 such that

(4.1) k�k.AkIt�E/ < " whenever k � k�:

For each t 2 .12; 1/ and " < minft � 1

2; 1� tg, there exists k� D k�."; t/ > 0 such

that

(4.2) k�k.AkIt�E1/ < " whenever k � k�:

Remark 4.3. In the previous results, we have used the open superlevel sets AkIt WD

fy W uk.y/ > tg. However, we could have used the closed superlevel sets AkIt WD

fy W uk.y/ � tg to obtain the same results. We also note that, for an arbitraryRadon measure !, we have

(4.3) !.AkIt / � !.AkIt / D !.@AkIt / D 0

for all but countably many t , for the reason that the family of sets f@AkIt W t 2 Rg

is pairwise disjoint and any Radon measure ! can assign positive values to only acountable number of such a family.

COROLLARY 4.4 For almost every t > 0, there exist ".t/ and k� D k�."; t/ > 0

such that

(i) k�k.AkIt�E/ D k�k. NAkIt�E/ < " if t 2 .0; 12/ and k � k�;

(ii) k�k.AkIt�E1/ D k�k. NAkIt�E

1/ < " if t 2 .12; 1/ and k � k�;

(iii) k�k.@AkIt�E/ D k�k.u�1k.t/�E/ < " if t 2 .0; 1

2/ and k � k�;

(iv) k�k.@Ak;t�E1/ D k�k.u�1

k.t/�E1/ < " if t 2 .1

2; 1/ and k � k�.

For the case t D 12

, only (i) and (iii) in Lemma 4.1 remain valid. To see this,we first show the following:

LEMMA 4.5 Let � be a Radon measure on RN such that � � HN�1. Let E be a

set of finite perimeter, and let uk be the mollification of �E . Then, for t D 12

and

" > 0, there exists k� D k�."/ such that

(4.4) k�k.E1 n AkI1=2/ < " and k�k.AkI1=2 nE/ < ":


PROOF: Since uk.y/ ! uE .y/ for HN�1-a.e. y, the dominated convergencetheorem implies that uk ! uE in L1.RN ; k�k/. Thus, given any " > 0, thereexists k� D k�."/ such that, when k � k�, we have

"

2�

Z

RN

juE � ukjdk�k

�

Z

E1nAkI1=2

.uE � uk/dk�k �

�

1 �1

2

�

k�k.E1 n AkI1=2/;

(4.5)

which implies

(4.6) k�k.E1 n AkI1=2/ � ":

In the same way, we compute

(4.7)"

2�

Z

AkI1=2nE

juE � ukjdk�k �

�

1

2� 0

�

k�k.AkI1=2 nE/;

which implies

(4.8) k�k.AkI1=2 nE/ � ":

�

The following remark shows that, with t D 12

and � D HN�1 @�E � 0, (ii)and (iv) in Lemma 4.1 do not hold.

Remark 4.6. If we define E WD fy 2 RN W jyj � 1g, then u�1

k.12/ � R

N n E forall k. Therefore, it is clear that

(4.9) HN�1..AkI1=2 nE1/ \ @�E/ D H

N�1.@�E/ ¹ 0 as k ! 1:

If we now define E WD fy 2 RN W jyj � 1g, then u�1

k.12/ � E for all k. Thus, we

have

(4.10) HN�1..E n AkI1=2/ \ @�E/ D H

N�1.@�E/ ¹ 0 as k ! 1:

LEMMA 4.7 There exists C < 1 such that, for all positive integers k and almost

all t 2 .0; 1/,

(4.11) HN�1.u�1k .t// � C:

PROOF: From Corollaries 4.2 and 4.4, it follows that, for almost all t 2 .0; 1/,the sequence of smoothly bounded sets AkIt D fuk > tg satisfies �AkIt

! �E1

�-a.e. if t 2 .12; 1/, or �AkIt

! �E�-a.e. if t 2 .0; 12/. Since � D HN�1

@�E � HN�1, it follows that �AkIt! �E everywhere except for a set of

Lebesgue measure zero.


We let V denote the Banach space C 1.K/ of vector fields on K endowedwith the norm

k k WD supy2K

�

j .y/j C

NX

iD1

jr i .y/j

�

;

where K is a compact set such that E b K. Let

TE . / WD

Z

E

.y/dy

and, for almost every t 2 .0; 1/, let

TkIt . / WD

Z

fyWuk.y/>tg

.y/dy D

Z

AkIt

.y/dy for 2 V :

Then, for 2 V , we define the linear operators

@TE . / WD TE .div / D

Z

E

div dy D �

Z

@�E

� � dHN�1

and

@TkIt . / WD TkIt .div / D

Z

AkIt

div dy D �

Z

@AkIt

� � dHN�1;

where � is the interior unit normal.Since u�1

k.t/ is a C1 manifold for almost every t , then

k@TkItk WD supk k�1

j@TkIt . /j D HN�1.u�1k .t//:

Indeed, with WD �k=j�kj defined on the manifold u�1k.t/, the norm-preserving

extension of to all of RN by Whitney’s extension theorem yields the inequality

k@TkItk WD supk k�1

j@TkIt . /j � HN�1.u�1

k .t//:

The opposite inequality is obvious.Moreover, we find by the dominated convergence theorem that

limk!1

@TkIt . / ! @TE . / for 2 V;

and thereforesupk

fj@TkIt . /jg < 1 for 2 V :

By the uniform boundedness principle (Theorem 2.2), we see that, since @TkIt is alinear functional on V whose weak limit @TE is independent of t , we have

supk

HN�1.u�1k .t// D sup

k

k@TkItk � C < 1;

where C > 0 is independent of t , which gives our desired result. �


The above argument simply rephrases the following basic fact from the theoryof currents. We know that, since E has finite perimeter, TE is an integral current.Moreover, the currents TkIt converge to TE weakly, and so do their boundaries,@TkIt ! @TE ; that is,

Z

u�1k.t/

� dHN�1 D

Z

@AkIt

� dHN�1 !

Z

@�E

� dHN�1

for each smooth differential .N � 1/-form � . Appealing to Theorem 2.2 yields ourresult.

LEMMA 4.8 Let u W � ! R be a Lipschitz function and E b � a set of finite

perimeter. Then

HN�1.@�E \ u�1.t// D 0 for almost all t :

PROOF: This result follows directly from Lemma 2.34, since we know thatHN�1.@�E/ < 1 for any bounded set of finite perimeter E b �. �

LEMMA 4.9 For almost every t 2 .12; 1/, we have

(4.12) HN�1.@�E \ u�1k .t// D 0

and

(4.13) limk!1

HN�1.@�E \ AkIt / D 0:

PROOF: This can be seen as follows. If we use Corollary 4.2 with� D HN�1

@�E, we obtain

limk!1

�.AkIt nE1/ D limk!1

HN�1.AkIt \ @�E/ D 0:

Clearly, (4.12) follows from Lemma 4.8 (see also Remark 4.3). �

Now we can establish the main theorem of this section.

THEOREM 4.10 (Approximation Theorem) For almost every t 2 .12; 1/, we have

limk!1

HN�1..E0 [ @�E/ \ u�1

k .t// D 0:

PROOF: Since the Lebesgue measure is absolutely continuous with respect toHN�1, then using (4.2) in Corollary 4.2 with s > 1

2leads to

jAkIs�E1j ! 0 as k ! 1:

Therefore, if we defineRkIs WD AkIs nE1;

it follows that

(4.14) jRkIsj ! 0 provided that s >1

2:


Remark 4.3 indicates that we have the option of defining AkIt WD fy W uk.y/ � tg

without altering the development. With this option in force, we have u�1k.s/ �

AkIs , and consequently, by the co-area formula (Theorem 2.32),Z

RkIs

jrukjdy D

Z 1

0

HN�1.u�1

k .t/ \RkIs/dt

D

Z 1

s

HN�1.u�1k .t/ \ .E0 [ @�E//dt:

Since ruk�* r�E and krukk1 � kr�Ek (Lemma 2.16), it follows from

Vitali’s convergence theorem for s > 12

thatZ

RkIs

jrukjdy ! 0:

Thus, for a subsequence if necessary, we can conclude that, for a.e. t > s,

HN�1.u�1

k .t/ \ .E0 [ @�E// ! 0 as k ! 1:

The dependence on the subsequence is illusory. The reason is that, if there werea subsequence such that, for a.e. t ,

HN�1.u�1

k .t/ \ .E0 [ @�E// ! ˛ ¤ 0 as k ! 1;

one could then appeal to our previous argument to conclude that, for some furthersubsequence and for a.e. t ,

HN�1.u�1k .t/ \ .E0 [ @�E// ! 0 as k ! 1;

which is contrary to our assertion that ˛ ¤ 0.Since s > 1

2is fixed arbitrarily at the beginning of the proof, we conclude that,

for a.e. t > 12

,

HN�1.u�1

k .t/ \ .E0 [ @�E// ! 0 as k ! 1:

�

5 Main Theorem

In this section we establish our main result of this paper, Theorem 5.2. LetF 2 DM1

loc.�/. We define, for almost every t , a measure �kIt for all Borel setsB b � by

(5.1) �kIt .B/ WD

Z

B\@AkIt

F .y/ � �.y/dHN�1.y/;

where F .y/ � �.y/ denotes the classical dot product of F with the unit normal �.We begin with a lemma that will lead to several of the assertions in Theorem 5.2.


LEMMA 5.1 If E b � is an arbitrary set of finite perimeter, then we have

(5.2)Z

E

F � ruk dy D

Z 1

0

Z

E\u�1k.t/

F � �k dHN�1 dt

for any F 2 L1loc.�/, where uk denotes the mollification of �E as introduced in

Definition 2.11 and Lemma 2.15.

PROOF: Let N be the set on which ruk D 0. ThenZ

E

F � ruk dy D

Z

EnN

F � ruk dy C

Z

N

F � ruk dy

D

Z

EnN

jrukjF � ruk

jrukjdy C 0

D

Z

E

jrukjg dy;

where g D�EnN F �ruk

jruk j. Then, by the co-area formula, we have

Z

E

F � ruk dy D

Z 1

0

Z

u�1k.t/\.EnN /

g dHN�1 dt

D

Z 1

0

Z

u�1k.t/\.EnN /

F � �k dHN�1 dt

D

Z 1

0

Z

u�1k.t/\E

F � �k dHN�1 dt;

where we have used �k.y/ D ruk.y/jruk.y/j

for y 2 u�1k.t/ \ .E n N /. �

With the help of Lemma 3.1 and the results in Section 4, we now establish ourmain theorem.

THEOREM 5.2 (Main Theorem) Let � � RN be an open set. Suppose that F 2

DM1loc.�/ with div F D � 2 M.�/. Let E b � be a set of finite perimeter.

Then

(i) For almost every s 2 .12; 1/, there exist a signed measure �i (independent

of s/ and a family of sets AkIs with smooth boundaries such that

(a) k�k.AkIs�E1/ ! 0I

(b) the measure �i is the weak-star limit of the measures �kIsI

(c) �i is carried by @�E in the sense that k�ik.� n @�E/ D 0;(d) k�ik � HN�1 @�EI


(e) limk!1 HN�1.@AkIs \ .E0 [ @�E// D 0I

(f) limk!1 k�kIsk.E0 [ @�E/ D 0;

(g) the density of �i , denoted as Fi � �, is called the interior normal tracerelative to E of F on @�E and satisfies

(5.3)Z

E1

div F DW �.E1/ D ��i .@�E/ D �

Z

@�E

.Fi � �/.y/dHN�1.y/I

(h) if .2F � ruk/�E is considered as a sequence of measures, then this

sequence converges weak-star to the measure .Fi � �/HN�1 @�E,

i.e.,

.2F � ruk/�E�* .Fi � �/HN�1 @�E in M.�/I

(i) k�ik D kFi � �k1I@�E;HN �1 � kF k1IE .

(ii) For almost every s 2 .0; 12/, there exist a signed measure �e (independent

of s) and a family of sets AkIs with smooth boundaries such that

(a) k�k.AkIs�E/ ! 0I

(b) the measure �e is the weak-star limit of �kIs;(c) �e is carried by @�E in the sense that k�ek.� n @�E/ D 0;(d) k�ek � HN�1 @�E;(e) limk!1 HN�1.@AkIs \E/ D limk!1 HN�1.u�1

k.s/ \E/ D 0;

(f) limk!1 k�kIsk.� nE0/ D limk!1 k�kIsk.E/ D 0I

(g) the density of �e , denoted as Fe � �, is called the exterior normal tracerelative to E of F on @�E and satisfies

(5.4)Z

E

div F DW �.E/ D ��e.@�E/ D �

Z

@�E

.Fe � �/.y/dHN�1.y/I

(h) if .2F � ruk/�E0 is considered as a sequence of measures, then this

sequence converges weak-star to the measure .Fe � �/HN�1 @�E,

i.e.,

.2F � ruk/�E0

�* .Fe � �/HN�1 @�E in M.�/I

(i) k�ek D kFe � �k1I@�E;HN �1 � kF k1I�nE .

PROOF: We will prove only part (i), since the proof of part (ii) is virtuallyidentical. For notational simplicity, we will use the notation � rather than �i inthe proof of part (i). Throughout the proof, we will consider only those values ofs 2 .1

2; 1/ for which u�1

k.s/ is a smooth manifold for all k, Lemma 3.1 holds, and

the results in Section 4 are valid for all the mollifications uk of �E , thus omittingat most a set S of measure zero. For the rest of the proof, fix s … S.

We start with (a). We consider the sets AkIs as in Lemma 4.1. The desiredresult follows directly from Corollary 4.2.


(b) Since F is bounded, Lemma 4.7 implies that there exists a constant C suchthat

(5.5) k�kIsk.�/ � C;

which yields, as in (3.8), the existence of a signed measure �s such that

(5.6) �kIs

�* �s in M.�/:

Utilizing (4.2), we also obtain that �.AkIs/ ! �.E1/. Since Lemma 3.1 yields�.AkIs/ D ��kIs.�/, we obtain, after letting k ! 1, that

(5.7) �.E1/ D ��s.�/:

Since the left side of equation (5.7) is independent of s, we show next that �sis also independent of s (and independent of the sequence in the weak-star con-vergence (5.6)). To see this, we fix any � 2 C 1c .�/ and note that, since F is adivergence-measure field, the product rule in Lemma 2.31 implies that �F is alsoa divergence-measure field. Proceeding as above with �F instead of F , we obtain

(5.8)Z

E1

div.�F / D �

Z

�

� d�s

for any � 2 C 1c .�/. Therefore, for any two measures �s and �s0 with limits asin (5.6), we have that

R

� � d�s DR

� � d�s0 for any � 2 C 1c .�/ and thus weconclude that �s D �s0 .

(c) Let A � � n @�E be an arbitrary Borel set. Referring to (2.8), we see that

kr�Ek.A/ D 0:

On the other hand, we know

(5.9)

0 D kr�Ek.A/

D inffkr�Ek.U / W A � U; U openg

D inffkr�Ek.U / W A � U; U open; kr�Ek.@U / D 0g:

In order to prove that k�k.A/ D 0, we proceed by contradiction by assumingk�k.A/ > 0. From (5.9), there is an open set U � A such that kr�Ek.@U / D 0

and

(5.10) kr�Ek.U / <k�k.A/

2kF k1:

From Lemma 5.1, we have

(5.11)Z

U

jF � rukjdy D

Z 1

0

Z

U\u�1k.t/

jF � �kjdHN�1 dt:


Since U is open and �kIt

�* � in M.�/,

k�k.A/ � 2

Z 1

12

k�k.U /dt � 2

Z 1

12

lim infk!1

k�kItk.U /dt

� 2

Z 1

0

lim infk!1

k�kItk.U /dt

� 2 lim infk!1

Z 1

0

k�kItk.U /dt

by Fatou’s lemma. Therefore, we have

k�k.A/ � 2 lim infk!1

Z 1

0

Z

u�1k.t/\U

jF .y/ � �.y/jdHN�1.y/dt

D 2 lim infk!1

Z

U

jF � rukjdy

� 2kF k1 limk!1

Z

U

jrukjdy

D 2kF k1 kr�Ek.U /

< k�k.A/;

where we have used Lemma 2.15(iv) and the fact that kr�Ek.@U / D 0. Thisyields a contradiction and thus establishes our result.

(d) Let A � @�E be a Borel set with HN�1.A/ D 0. Then, appealing to (2.8),we find that kr�Ek.A/ D 0. From this, the proof can proceed precisely as in (c)to yield our desired conclusion.

(e) This is the result of Theorem 4.10.

(f) In view of the definition

�kIs.B/ WD

Z

@AkIs\B

F .y/ � �.y/dHN�1.y/

and the fact that F is bounded, the result follows immediately from (e).

(g) From (a), we have the existence of smoothly bounded sets such that

(5.12) k�k.AkIs�E1/ ! 0 as k ! 1;


where s > 12

is fixed as in the beginning of the proof. From Lemma 3.1, we knowthat our desired result holds for the sets Ak;s:

(5.13) �.AkIs/ WD

Z

AkIs

div F D �

Z

@AkIs

F .y/ � �.y/dHN�1.y/:

We note that, with our notation in force, we may write (5.13) as

(5.14) �.AkIs/ D ��kIs.�/ D ��kIs.@AkIs/:

Since

(5.15) �.AkIs/ ! �.E1/ and �kIs.�/ ! �.�/ as k ! 1;

we obtain

�.E1/ D ��.@�E/:

Because k�k � HN�1 @�E, we know that there exists Fi � � 2 L1.@�E/ suchthat

�.B/ D

Z

B\@�E

.Fi � �/.y/dHN�1.y/;

which gives (5.3).

(h) From Lemma 5.1, we obtain

limk!1

Z

E

F � ruk dy D limk!1

Z 1

0

Z

u�1k.t/\E

F .y/ � �k.y/dHN�1.y/dt

D limk!1

Z 1

0

�kIt .E/dt:

Thus,

�EF � ruE .�/ WD limk!1

Z

�

�EF � ruk dy

D limk!1

Z

E

F � ruk dy

D limk!1

Z 1

12

�kIt .E/dt C limk!1

Z 1

12

�kIt .E0/dt (by (f) above)

D limk!1

Z 1

12

�kIt .�/dt

D1

2�.�/:


Let ' be a function in C 1c .�/. Since 'F is also a bounded divergence-measurefield, we can proceed as above with the vector field 'F instead of F to concludethat

(5.16)Z

�

' d �EF � ruE D1

2

Z

�

' d�;

which implies that � D 2�EF � ruE .

(i) We have that, for HN�1-a.e. y 2 @�E,

(5.17) Fi � �.y/ D limr!0

�.B.y; r//

kr�Ek.B.y; r//;

where we can choose the balls B.y; r/ such that

kr�[email protected]; r// D k�[email protected]; r// D 0:

Using a similar argument as in (c), we obtain

k�k.B.y; r//

D 2

Z 1

12

k�k.B.y; r//dt

D 2 limk!1

Z 1

12

k�kItk.B.y; r//dt

D 2 limk!1

Z 1

12

�

k�kItk.B.y; r/ \E1/C k�kItk.B.y; r/ \ .E0 [ @�E//�

dt

D 2 limk!1

Z 1

12

k�kItk.B.y; r/ \E1/dt

D 2 limk!1

Z 1

12

Z

u�1k.t/\B.y;r/\E1

jF � �jdHN�1 dt

� 2kF k1IE1 limk!1

Z 1

12

Z

u�1k.t/\B.y;r/

dHN�1 dt;

where, in proceeding from the third equality to the fourth, we have used the factthat k�kItk.E

0 [ @�E/ ! 0 as k ! 1 for a.e. t > 12

.Therefore, from (5.17), we obtain

jFi � �.y/j � limr!0

k�k.B.y; r//

kr�Ek.B.y; r//

� 2kF k1IE1 limr!0

limk!1

R 11=2

R

u�1k.t/\B.y;r/ dH

N�1 dtR

B.y;r/ jrukjD


D 2kF k1IE1 limr!0

limk!1

R 11=2

R

u�1k.t/\B.y;r/ dH

N�1 dtR 10

R

u�1k.t/\B.x;r/ dH

N�1 dt

D kF k1IE1 D kF k1IE : �

As a direct result, we obtain the Gauss-Green theorem for divergence-measurefields over sets of finite perimeter. This also shows that our definition of the normaltrace is in agreement with that given in the sense of distributions, Definition 2.19.

THEOREM 5.3 (Gauss-Green Theorem) Let � � RN be an open set. Let F 2

DM1loc.�/ and let E b � be a bounded set of finite perimeter. Then,

(5.18)Z

E1

' div F C

Z

E1

F � r' D �

Z

@�E

'.Fi � �/dHN�1

for all ' 2 C1c .�/, where Fi � � is the interior normal trace of F relative to E on

@�E.

PROOF: From Theorem 2.31, it follows that 'F is a bounded divergence-measure field and

(5.19) div.'F / D ' div F C F � r':

Following the proof of Theorem 5.2 applied to 'F (instead of F ), we obtain

(5.20)Z

E1

div.'F / D �

Z

@�E

' .Fi � �/dHN�1;

which, due to (5.19), gives the desired result. �

We conclude this section with the following remark.

Remark 5.4. Theorem 5.3 implies that, when E is an open set of finite perimeter,our trace Fi � � agrees with the one defined in (2.10).

6 The Divergence Measure of Jump Sets via the Normal Trace

In Theorem 5.2, we have defined the interior and exterior normal traces of F 2

DM1loc.�/, Fi � � and Fe � �, over the boundary of a set of finite perimeter E b �.

In order to obtain the interior normal trace of F on @� QE, where QE WD E0 [ @mE,we reproduce the proof of Theorem 5.2 and apply it to QE. Therefore, the tracemeasure, denoted by ��, is obtained by using the level sets BkIs D fvk > sg forsome s 2 .1

2; 1/, where vk is the mollification of � QE

. We note that, for all y 2 �,

�" � �E .y/C �" � � QE.y/ D 1;

and thereforev�1k .s/ D u�1

k .1 � s/;


where 1 � s 2 .0; 12/. Since �� is the interior unit normal to QE, we have

��.�/ D � limk!1

Z

@BkIs

F � � dHN�1 D � limk!1

Z

@AkI1�s

F � � dHN�1

D � limk!1

�kI1�s.RN / D ��e.�/:

(6.1)

The following observation now becomes evident.

COROLLARY 6.1 The interior trace of F relative to QE on @�E is the same as

minus the exterior trace of F relative to E on @�E.

In order to establish the relation between �i and �e , we subtract (5.4) from (5.3)and obtain the following formula for � D div F .

COROLLARY 6.2 �.@�E/ D

Z

@�E

.Fi � � � Fe � �/.y/dHN�1.y/.

We offer the following simple example to illustrate our result. This examplealso dramatically demonstrates the difference between the classical derivative andthe weak (distributional) derivative.

EXAMPLE 6.3 Consider the most elementary situation: N D 1, � WD .�1; 2/,E WD Œ0; 1�, and f is a nondecreasing function defined on .�1; 2/ that is contin-uous everywhere except at y D 0; 1, at which points we assume that f is right-continuous.

(i) Case 12< s < 1. Since f is in BV, we know that f 0 D � for some

measure �. Then, according to Theorem 5.2,

�.E1/ D �..0; 1// WD

Z 1�

0C

f 0 D Fi � �.1/ � Fi � �.0/;

where Fi � �.1/ D limy!1� f .y/ and Fi � �.0/ D f .0C/. Indeed, the sets AkIs ,with fixed s 2 .1

2; 1/, form a nested family of open intervals contained in Œ0; 1�.

The measures �k correspond to f evaluated on the point masses located at yk;thus, as in (5.15), f .yk/ converges to a limit, Fi � �.1/.

(ii) Case 0 < s < 12

. Then the sets AkIs , s 2 .0; 12/, form a nested family of

open intervals containing Œ0; 1�. Similar to the above, we have

�.E/ D �.Œ0; 1�/ WD

Z 1C

0�

f 0 D Fe � �.1/ � Fe � �.0/;

the measures �k correspond to f evaluated on the point masses located at yk , andthus Fe � �.1/ D f .1C/ and Fe � �.0/ D limy!0� f .y/.


7 Consistency of the Normal Trace with the Classical Trace

We now proceed to show the consistency of our normal trace with the classicaltrace when F is continuous. First we have the following lemma:

LEMMA 7.1 Let � D div F for F 2 DM1loc.R

N / \ C.RN I RN /. Then

k�k.G/ D 0

for any set G that can be written as the graph of a Lipschitz function f .

PROOF: First we have

G WD f.y0; f .y0// W y0 2 W � RN�1g:

By regularity of �, it suffices to show that �.K/ D 0 for any compact setK � G. Given such K � G, let Uk � R

N be a sequence of open sets satisfying

(7.1) k�k.Uk/ ! k�k.K/:

Fix any set Uk . We note by Besicovitch’s theorem that Uk can be written up to aset of k�k-measure zero as a countable union of disjoint open parallelepipeds I ki(the fact that we can use parallelepipeds instead of balls follows from Morse [53]).Thus, we have

(7.2)1[

iD1

I ki � Uk and k�k

�

Uk n

1[

iD1

I ki

�

D 0:

Denote Uk simply as U and I ki as Ii . We fix an i and note that, for t small enough,the graphs Tt WD f.y0; f .y0/C t / W y0 2 W � R

N�1g and Bt WD f.y0; f .y0/� t / W

y0 2 W � RN�1g are contained in Ii . Let Rt be the region inside Ii , bounded

above and below by Tt and Bt , respectively. We define

˛t D

Z

@Rt n.Tt [Bt /

F .y/ � �.y/dHN�1.y/ for a.e. t ;

where �.y/ is the interior unit normal to Rt on @Rt n .Tt [Bt /. Since Lemma 3.1applies to Rt for a.e. t , we arrive at

�.Rt / D

Z

Rt

div F

D �

Z

Bt

F .y/ � �.y/dHN�1.y/ �

Z

Tt

F .y/ � �.y/ dHN�1.y/ � ˛t D


D

Z

Tt

F .y0; yn � 2t/ � �.y/dHN�1.y/

�

Z

Tt

F .y0; yn/ � �.y/dHN�1.y/ � ˛t

D

Z

Tt

.F .y0; yn � 2t/ � F .y0; yn// � �.y/dHN�1.y/ � ˛t :

Since F is continuous and ˛t ! 0 as t ! 0, we find that there exists t0.";F ; G/ >0 such that

�.Rt / � " for all t � t0.";F ; G/:

Then we have

�.Ii \G/ D limt!0

�.Rt / � ";

which implies �.Ii \G/ D 0 since " is arbitrary. Therefore, using (7.1) and (7.2),we obtain

�.K/ D limk!1

X

�.I ki \G/ D 0:

�

THEOREM 7.2 If F 2 DM1loc.�/ is continuous and E b � is a set of finite

perimeter, then �i D .Fi � �/HN�1 @�E D F � ruE , where F � ruE is the

weak-star limit of the measures F � ruk . Moreover, the normal trace Fi � � is in

fact the classical dot product F ��, where � is the measure-theoretical interior unit

normal to E on @�E.

PROOF: We recall that, by definition, E D E1[@�E. Denote QE D E0[@�E.Then we have

F � ruE D limk!1

Z

RN

F � ruk dy

D limk!1

Z

RN

�EF � ruk dy C limk!1

Z

RN

� QEF � ruk dy:

If vk denotes the convolution � QE� �1=k , since uk C vk D 1, we obtain

F � ruE D limk!1

Z

RN

�EF � ruk dy � limk!1

Z

RN

� QEF � rvk dy

D�i

2C�e

2D1

2.�i C �i � �.@�E// D �i �

1

2�.@�E/;

where we have used Theorem 5.2(h) and Corollary 6.2.


Since @�E is an .N � 1/-rectifiable set (see (2.7)), it follows from Lemma 7.1that

k�k.@�E/ D 0;

that is,F � ruE D �i :

Thus, for HN�1-a.e. y 2 @�E,

.Fi � �/.y/ D limr!0

F � ruE .B.y; r//

kr�Ek.B.y; r//D limr!0

limk!1

R

B.y;r/ F � ruk dxR

B.y;r/ dkr�Ek:

Since ruk ! r�E weak-star and F is continuous, and noting that rj can bechosen such that kr�[email protected]; rj // D 0, we obtain

.Fi � �/.y/ D limj!1

R

B.y;rj /F � r�E

R

B.y;rj /dkr�Ek

D limj!1

R

B.y;rj /F .x/ � �.x/ dkr�Ek.x/

R

B.y;rj /dkr�Ek.x/

D F .y/ � �.y/;

by differentiation of measures. �

The following corollary gives more information of the trace �i and the levelsets u�1

k.s/ when s ! 1

2C.

COROLLARY 7.3 The trace measure �i given in Theorem 5.2 satisfies

�i .RN / D 2 lim

k!1

Z

E

F � ruk dy D 2 limk!1

lims!1=2C

Z

AkIs

F � ruk dy

D 2 limk!1

Z

AkI1=2

F � ruk dy:

PROOF: Theorem 5.2 (h) shows

(7.3) �i .RN / D 2 lim

k!1

Z

E

F � ruk dy:

Using Lemma 5.1, we find

�i .RN / D 2 lim

s!1=2C

Z 1

s

�i .RN /dt D 2 lim

s!1=2Climk!1

Z 1

s

Z

u�1k.t/

F � � dHN�1 dt

D 2 lims!1=2C

limk!1

Z

AkIs

F � ruk dy:


One can easily verify that the limits s ! 12C and k ! 1 can be interchanged.

Noting thatS

s>1=2AkIs D AkI1=2; we conclude

�i .RN / D 2 lim

k!1lim

s!1=2C

Z

AkIs

F � ruk dy D 2 limk!1

Z

AkI1=2

F � ruk dy:

�

8 One-Sided Approximation of Sets of Finite Perimeter

It is well-known that a set of finite perimeter E cannot be approximated bysmooth sets that lie completely in the interior of E. For example, consider theopen unit disk with a single radius removed, and let U be the resulting open set.Then the Hausdorff measure of the boundary of U is 2� plus the measure of theradius, while the Hausdorff measure of the reduced boundary is 2� . Thus, if Ukis an approximating open subset of U , then its boundary will be close to that of Uand so its Hausdorff measure will be close to 2� plus 1. Adding more radii, saym of them, will force the approximating set to have boundaries whose Hausdorffmeasure is close to 2� plus m. In general, if we let K denote any compact subsetwithout interior and of infinite Hausdorff measure, then the approximating sets willhave boundaries whose measures will necessarily tend to infinity.

On the other hand, we have seen (Theorem 3.3) that one-sided approximationis possible for open sets of class C 1.

We have the following:

PROPOSITION 8.1 Let U � RN be an open set with HN�1.@U / < 1. Then there

exists a sequence of bounded open sets Uk � Uk � U such that

(i) jUkj D jUkj;(ii) jUkj ! jU j;

(iii) HN�1.@Uk/ ! HN�1.@U /.

PROOF: By definition, for each integer k, there exists a covering of @U by balls

@U �[

Bi .ri /;

each with radius ri , such that1

X

iD1

HN�1.@Bi .ri // D

1X

iD1

!N�1rN�1i < H

N�1.@U /C1

k;

where !N�1 is the HN�1 measure of the boundary of the unit ball in RN . Since

@U is compact, the covering may be taken as a finite covering, say by m of them,B1.r1/; : : : , Bm.rm/. Then the open set Vk WD

SmiD1Bi .ri / has the property that

@Vk �

m[

iD1

@Bi .ri /


and therefore that

HN�1.@Vk/ � H

N�1

� m[

iD1

@Bi .ri /

�

�

1X

iD1

!N�1rN�1i < H

N�1.@U /C1

k:

Thus, the open sets Uk WD U n V k � U satisfy our desired result. Note that theyare not smooth. �

Given an arbitrary set of finite perimeter E, we know from Section 4 that Ecan be approximated by sets with smooth boundaries essentially from the measure-theoretic interior ofE, that is, a one-sided approximation can “almost” be achieved(see Theorem 5.2(e)). On the other hand, the next result shows that, if E is suffi-ciently regular, there does, in fact, exist a one-sided approximation. The conditionof regularity we impose is similar to Lewis’s uniformly flat condition in potentialtheory [49].

THEOREM 8.2 Suppose thatE is a bounded set of finite perimeter with the property

that, for all y 2 @E, there are positive constants c0 and r0 such that

(8.1)jE0 \ B.y; r/j

jB.y; r/j� c0 for all r � r0:

Then there exists t 2 .0; 1/ such that

(8.2) AkIt b E for large k:

PROOF: Choose a mollifying kernel � such that � D 1 on B.0; 12/. If y 2 @E,

we have

vk.y/ WD �RN nE � �"k.y/ D

1

"Nk

Z

B.y;"k/

�RN nE .x/�

�

x � y

"k

�

dx

�1

"Nk

Z

B.y;"k=2/

�RN nE .x/dx

Dj.RN nE/ \ B.y; "k=2/j

"Nk

DjE0 \ B.y; "k=2/j

"Nk

�c0

2NWD Qc0;

where 0 < Qc0 < 1 depends only on the dimension N and is independent of thepoint y. Note that uk.y/C vk.y/ D 1 for all y 2 R

N . Therefore, for all y 2 @E,

uk.y/ D 1 � vk.y/ � 1 � Qc0:

Thus, taking 1 � Qc0 < t < 1, we see that AkIt \ @E D ¿. Consequently, eachconnected component of the open set AkIt lies either in the interior of E or in itsexterior and thus must lie in its interior. �


COROLLARY 8.3 LetE be a bounded set of finite perimeter with uniform Lipschitz

boundary. Then there exists T 2 .0; 1/ such that AkIT b E.

PROOF: Since E has a uniform Lipschitz boundary, for each x 2 @E, thereis a finite cone Cx with vertex x that completely lies in the complement of E.Each cone Cx is assumed to be congruent to a fixed cone C . This implies that thehypothesis of Theorem 8.2 is satisfied. Therefore, there exists 0 < T < 1 such thatuk.y/ < T for all k and all y 2 @E. �

DEFINITION 8.4 An open set U � RN is called an extension domain for F 2

DM1.U / if there exists a field F� 2 DM1.RN / such that F D F

� on U .

THEOREM 8.5 An open set U satisfying HN�1.@U / < 1 is an extension domain

for any F 2 DM1.U /. More generally, if F 2 DM1loc.�/, then any open set of

finite perimeter U b � is an extension domain for F .

PROOF: We define an extension of F by

F�.y/ WD �U .y/F .y/ for all y 2 R

N .

According to Definition 2.18, it suffices to show that

sup

� Z

RN

F� � r' W j'j � 1; ' 2 C1

c .RN /

�

< 1:

We consider first the case HN�1.U / < 1. Let Uk be the sequence of approxi-mate sets given in Proposition 8.1. Therefore, for any ' 2 C1

c .RN / with j'j � 1,we employ our general Gauss-Green theorem, Theorem 5.2, to obtain

Z

Uk

F � r' dy C

Z

Uk

' div F D �

Z

@Uk

' Fi � � dHN�1:

Thus,Z

Uk

F � r' dy D �

Z

Uk

' div F �

Z

@Uk

' Fi � � dHN�1

� kdiv F k.Uk/C kF k1HN�1.@Uk/

� kdiv F k.U /C kF k1HN�1.@Uk/:

Letting k ! 1, we obtainZ

U

F � r' dy � kdiv F k.U /C kF k1HN�1.@U / < 1:


Thus,Z

RN

F� � r' dy D

Z

U

F � r' dy < 1:

We now consider the case that U b � is a set of finite perimeter and F 2

DM1loc.�/. Proceeding as above and using Theorem 5.2,

Z

RN

F� � r' dy D

Z

U

F � r' dy

D �

Z

U

' div F �

Z

@�U

' Fi � � dHN�1

� kdiv F k.U /C kF k1HN�1.@�U/ < 1:

�

COROLLARY 8.6 Let U � RN be a bounded and open set with HN�1.@U / < 1.

Let F 1 2 DM1.U / and F 2 2 DM1.RN n U/: Then, with

F .y/ WD

(

F 1.y/; y 2 U ;

F 2.y/; y 2 RN n U ,

we have

F 2 DM1.RN /:

PROOF: Applying the previous result to

F�1 WD �U .y/F 1.y/ for all y 2 R

N

and

F�2 WD �

RN nUF 2.y/ for all y 2 RN ;

we see thatF D F

�1 C F

�2 :

�

9 Cauchy Fluxes and Divergence-Measure Fields

The physical principle of balance law of the form

(9.1)Z

@E

f .y; �.y//dHN�1.y/C

Z

E

b.y/dy D 0

is basic in all of continuum physics. Here, �.y/ is the interior unit normal to theboundary @E of E. In mechanics, f represents the surface force per unit areaon @E, while in thermodynamics, f gives the heat flow per unit area across theboundary @E.


In 1823, Cauchy [12] (also see [13]) established the stress theorem that is prob-ably the most important result in continuum mechanics: If f .y; �.y//, defined foreach y in an open region � and every unit vector �, is continuous in y and b.y/ isuniformly bounded on �, and if (9.1) is satisfied for every smooth region E b �,then f .y; �/ must be linear in �. The Cauchy postulate states that the densityflux f through a surface depends on the surface solely through the normal at thatpoint. For instance, if f .y; �/ represents the heat flow, then the stress theorem

states that there exists a vector field F such that

f .y; �/ D F .y/ � �:

Since the time of Cauchy’s stress result, many efforts have been made to gener-alize his ideas and remove some of his hypotheses. The first results in this directionwere obtained by Noll [54] in 1959, who set up a basis for an axiomatic founda-tion for continuum thermodynamics. In particular, Noll showed that the Cauchypostulate may directly follow from the balance law. In [42], Gurtin and Martins in-troduced the concept of Cauchy flux and removed the continuity assumption on f .In [71], Ziemer proved Noll’s theorem in the context of geometric measure theory,in which the Cauchy flux was first formulated at the level of generality with sets offinite perimeter in the absence of jump surfaces, “shock waves.”

However, as we explain below, all the previous formulations of (9.1) do notallow the presence of “shock waves”; one of our main intentions in this paper is todevelop a theory that allows the presence of “shock waves.”

In this section we first introduce a class of Cauchy fluxes that allows the pres-ence of the exceptional surfaces or “shock waves,” and we then prove that sucha Cauchy flux induces a bounded divergence-measure (vector) field F so that theCauchy flux over every oriented surface can be recovered through F and the normalto the oriented surface. Before introducing this framework, we need the followingdefinitions.

DEFINITION 9.1 An oriented surface in� is a pair .S; �/ so that S b � is a Borelset and � W R

N ! SN�1 is a Borel measurable unit vector field that satisfy the

following property: There is a set E b � of finite perimeter such that S � @�E

and

�.y/ D �E .y/�S .y/;

where �S is the characteristic function of the set S and �E .y/ is the interiormeasure-theoretic unit normal to E at y.

DEFINITION 9.2 Two oriented surfaces .Sj ; �j /; j D 1; 2, are said to be compat-

ible if there exists a set of finite perimeter E such that Sj � @�E and �j .y/ D

�E .y/�Sj.y/, j D 1; 2: For simplicity, we will denote the pair .S; �/ simply as

S , with the implicit understanding that S is oriented by the interior normal of somesetE of finite perimeter. We define �S D .S;��/, which is regarded as a differentoriented surface.


DEFINITION 9.3 Let � be a bounded open set. A Cauchy flux is a functional Fthat assigns to each oriented surface S WD .S; �/ b � a real number and has thefollowing properties:

(i) F.S1[S2/ D F.S1/CF.S2/ for any pair of compatible disjoint surfacesS1; S2 b �.

(ii) There exists a nonnegative Radon measure � in � such that

jF.@�E/j � �.E1/

for every set of finite perimeter E b �.(iii) There exists a constant C such that

jF.S/j � CHN�1.S/

for every oriented surface S b �.

This general framework for Cauchy fluxes allows the presence of exceptionalsurfaces, “shock waves,” in the formulation of the axioms, on which the measure �has support. On these exceptional surfaces, the Cauchy flux F has a discontinuity,i.e., F.S/ ¤ �F.�S/. In fact, the exceptional surfaces are supported on thesingular part of the measure � in general. When � reduces to the N -dimensionalLebesgue measure LN , the formulation reduces to Ziemer’s formulation in [71],and in this case � vanishes on any HN�1-dimensional surface, which excludesshock waves.

The theory developed in this paper allows us to approximate the exceptionaloriented surfaces or “shock waves” with smooth boundaries and rigorously passto the limit to recover the flux on the exceptional oriented surfaces as the pre-cise representative. This allows us to capture measure production density in theformulation of the balance law and entropy dissipation for entropy solutions ofhyperbolic conservation laws. Once we know the flux across every surface, weproceed to obtain a rigorous derivation of nonlinear systems of balance laws withmeasure source terms from the physical principle of balance law in Section 10. Theframework also allows the recovery of Cauchy entropy fluxes through the Lax en-tropy inequality for entropy solutions of hyperbolic conservation laws by capturingentropy dissipation; see Section 11.

The main theorem of this section is the following:

THEOREM 9.4 Let F be a Cauchy flux in�. Then there exists a unique divergence-

measure field F 2 DM1loc.�/ such that

(9.2) F.S/ D �

Z

S

Fi � � dHN�1

for every oriented surface .S; �/ b �, where Fi � � is the normal trace of F to the

oriented surface.


When � reduces to the N -dimensional Lebesgue measure LN , as in Ziemer’sformulation, the vector field F satisfies div F 2 L1 and F.S/ D �F.�S/ forevery surface S , which thus excludes shock waves where the Cauchy flux F has adiscontinuity, i.e., F.S/ ¤ �F.�S/.

In order to establish Theorem 9.4, we need Lemma 9.6, which was first shownin Degiovanni, Marzocchi, and Musesti [26]. Here we offer a simplified proofof this fact for completeness. In particular, Lemma 9.6 is a direct application ofTheorem 9.5 (due to Fuglede) below. We also refer to Schuricht [60] for a differentapproach in formulating the axioms in Definition 9.3. Our Theorem 9.4 followsthen by an approximation and Theorem 5.2.

The following theorem, due to Fuglede, is a generalization of Riesz’s theorem,whose proof can be found in [38].

THEOREM 9.5 Let � be a nonnegative measure defined on a �-field V of subsets

of a fixed set X and X 2 V . Let ' be an additive set function defined on a system

of sets U � V such that all finite unions of disjoint sets from U , together with the

empty set, form a field F that generates V . Assume that �.A/ < 1 for every

A � U . Then there exists a function g.y/ 2 L1.X;V; �/ with the property that

'.A/ D

Z

A

g.y/d� for every A 2 U

if and only if the following hold:

(i) For every " > 0, there exists ı > 0 such thatPniD1 j'.Ai /j � " for every

finite system of disjoint sets A1; : : : ; An from U wherePniD1 �.Ai / < ı.

(ii) There is a finite constant C such thatPniD1 j'.Ai /j � C for every finite

system of disjoint sets A1; : : : ; An from U .

The function g is then essentially uniquely determined. Under the additional as-

sumption that �.X/ < 1, condition (ii) is a consequence of condition (i).

Let fI g be the collection of all closed cubes in RN of the form

I D Œa1; b1� � � � � � ŒaN ; bN �;

where a1; b1; : : : ; aN ; bN are real numbers. For almost every �j 2 Œaj ; bj �, wedefine

I�j D fy 2 I W yj D �j g:

We define the vectors e1; : : : ; eN so that the j th component of ej is �1 and theother components are 0. We orient the surface I�j with the vector ej .

LEMMA 9.6 Let F be a Cauchy flux in �. Then there exists a divergence-measure

field F 2 DM1loc.�/ such that, for every cube I D Œa1; b1� � � � � � ŒaN ; bN � b �

and almost every �j 2 Œaj ; bj �,

F.I�j / D �

Z

I�j

F .y/ � ej dHN�1.y/:


PROOF:

Step 1. We fix j 2 f1; : : : ; N g. For every cube I � �, we define

�j .I / D

Z bj

aj

F.I�j /d�j :

We have

(9.3) j�j .I /j �

Z bj

aj

jF.I�j /jd�j � C

Z bj

aj

Z

I�j

dHN�1 d�j D C jI j:

Thus, from Theorem 9.5, there exists a function f j 2 L1.�/ such that

�j .I / D

Z

I

f j dy for every I:

In fact, inequality (9.3) implies that f j 2 L1.�/ since, for LN -a.e. y,

f j .y/ D limjI j!0y2I

R

I fj dx

jI j� C:

Fubini’s theorem implies that

(9.4) �j .I / D

Z bj

aj

F.I�j /d�j D

Z

I

f j dy D

Z bj

aj

Z

I�j

f jdHN�1 d�j :

Let �j 2 Œaj ; bj �, ˛k;j , and ˇk;j be sequences such that

˛k;j � �j � ˇk;j ;

where ˛k;j is an increasing sequence that converges to �j as k ! 1, and ˇk;j is adecreasing sequence that converges to �j as k ! 1. Thus, from (9.4), we obtain

(9.5)1

˛k;j � ˇk;j

Z bj

aj

F.I�j /d�j D1

˛k;j � ˇk;j

Z bj

aj

Z

I�j

f j dHN�1 d�j :

We let k ! 1 to obtain that, for a.e. �j ,

(9.6) F.I�j / D

Z

I�j

f j dHN�1:

DefineF WD .f 1; : : : ; f N /:

Then we find, for almost every �j , j 2 f1; : : : ; N g,

(9.7) F.I�j / D �

Z

I�j

F .y/ � ej dHN�1.y/:


Step 2. We now prove that the divergence of F , in the sense of distributions, isa measure. We define, for a.e. cube I ,

(9.8) �.I / WD �

Z

@I

F .y/ � �.y/dHN�1.y/;

where � is the interior unit normal to @I . From Step 1 and the definition of Cauchyfluxes, we have

(9.9) j�.I /j D jF.@I /j � �.I 1/ � �.I /

for almost all closed cubes. Thus, we can again apply Theorem 9.5 to concludethat there exists a function g 2 L1.�I �/, uniquely defined in � up to a set of�-measure zero, such that

(9.10) �.I / D �

Z

@I

F .y/ � �.y/dHN�1.y/ D

Z

I

g.y/d�

for almost every closed cube I � �.Denote by Q� the measure given by g d� in �. We now prove

(9.11) div F D Q�

in the sense of distributions in any open set U b �.Let I b U be any closed cube. Then, for any � 2 C 1 with support contained

in I ,

(9.12)Z

U

F � r� dy D lim"!0

Z

U

F " � r� dy D � lim"!0

Z

U

� div F " dy;

where F " D F � �" and � is the standard mollifying kernel. We now prove that,for LN -a.e. y 2 U ,

Q�".y/ D divF ".y/;

where Q�" is the convolution of function �" with the measure Q� ; that is,

(9.13) Q�".y/ WD .�" � Q�/.y/ D

Z

�

�".y � x/d Q�.x/:

From (9.9)–(9.10), we find that, for " < dist.@U; @�/,Z

I

div F ".y/dy D �

Z

@I

F ".y/ � �.y/dy

D �

Z

@I

Z

RN

F .y � x/ � �.y/�".x/dx dy

D �

Z

RN

Z

@I

F .y � x/ � �.y/�".x/dy dx D


D �

Z

RN

� Z

@Ix

F .y/ � �.y/dy

�

�".x/dx

D

Z

RN

Q�.Ix/�".x/dx;

where Ix D fy W ai � yi � xi � bi ; i D 1; : : : ; N g. We can consider the smoothfunction �" as a measure in R

N , say �", by defining �".A/ DR

A �".x/dx for anyBorel set A. We can also extend the measure Q� by zero outside �. Therefore, wefind

(9.14)Z

RN

Q�.Ix/�".x/dx D . Q� � �"/.I / D .�" � Q�/.I / D

Z

RN

�".Ix/d Q�.x/:

From (9.14) and using (9.13), we computeZ

RN

�".Ix/d Q�.x/ D

Z

�

�".Ix/d Q�.x/ D

Z

�

� Z

Ix

�".y/dy

�

d Q�.x/

D

Z

�

Z

I

�".y � x/dy d Q�.x/

D

Z

I

� Z

�

�".y � x/d Q�.x/

�

dy

D

Z

I

.�" � Q�/.y/dy

D

Z

I

Q�".y/dy:

Therefore,Z

I

div F ".y/dy D

Z

I

Q�".y/dy:

Since the cube I b U is arbitrary, this shows that Q�".y/ D div F ".y/ for LN -a.e. y 2 U . Using this in (9.12), we obtain

Z

U

F � r� dy D � lim"!0

Z

U

� div F " dy

D � lim"!0

Z

U

� Q�" dy D �

Z

U

�.y/d Q�.y/;

(9.15)

since the sequence of measures Q�" converges locally weak-star to Q� in� as " ! 0.

�


PROOF OF THEOREM 9.4: Using Lemma 9.6, it follows that there exists anF 2 DM1

loc.�/ such that, for any cube I D Œa1; b1� � � � � � ŒaN ; bN � b �,

(9.16) F.I�j / D �

Z

I�j

F .y/ � ej dHN�1.y/

for almost every �j 2 Œaj ; bj �.Let .S; �/ be an oriented surface. Then there exists a set of finite perimeter E

such that S � @�E. We approximate S with closed cubes such that

(9.17) S D

1\

iD1

Ji ;

where each Ji is a finite union of closed cubes (which can be chosen so that (9.16)holds for �j D aj and � D bj ) centered at S and JiC1 � Ji . Using Lemma 2.10,we have @�.Ji \ E/ � .@�E \ Ji / [ .@�Ji \ E/ [ .@�E \ @�Ji /. From (9.17)and the fact that the cubes can also be chosen so that HN�1.@�E \ @�Ji / D 0, wehave

(9.18) Œ@�.Ji \E/��ŒS [ .@�Ji \E/� D Ni ;

where limi!1 HN�1.Ni / D 0. Since F.Ni / � CHN�1.Ni /, we obtain

(9.19) limi!1

F.Ni / D 0:

The definition of Cauchy fluxes implies that

jF.@�.Ji \E//j � �..Ji \E/1/:

The standard measure theory and (9.17) imply that

(9.20) limi!1

�..Ji \E/1/ D ��

\

Ji \E1�

D �.S \E1/ D 0;

since S � @�E. Therefore, we have

(9.21) limi!1

jF.@�.Ji \E//j D 0:

On the other hand, using Theorem 5.2, we have

(9.22)

ˇ

ˇ

ˇ

ˇ

�

Z

@�.Ji \E/

Fi � � dHN�1

ˇ

ˇ

ˇ

ˇ

D

ˇ

ˇ

ˇ

ˇ

Z

.Ji \E/1

div F

ˇ

ˇ

ˇ

ˇ

� kdiv F k..Ji \E/1/;

which yields (from (9.20) with kdiv F k instead of �):

(9.23) limi!1

ˇ

ˇ

ˇ

ˇ

Z

@�.Ji \E/

Fi � � dHN�1

ˇ

ˇ

ˇ

ˇ

D 0:


Using (9.18), (9.21), and (9.16), we obtain

(9.24)

limi!1

jF.@�.Ji \E//j

D limi!1

jF.S/C F.@�Ji \E/j

D limi!1

ˇ

ˇ

ˇ

ˇ

F.S/ �

Z

@�Ji \E

F .y/ � �.y/dHN�1

ˇ

ˇ

ˇ

ˇ

D 0:

From (9.18), (9.19), and (9.23), we have

(9.25) limi!1

ˇ

ˇ

ˇ

ˇ

�

Z

@�Ji \E

F .y/ � �.y/dHN�1 �

Z

S

Fi � � dHN�1

ˇ

ˇ

ˇ

ˇ

D 0:

Combining (9.24) with (9.25) yields

F.S/ D �

Z

S

Fi � � dHN�1:

Assume now that there exists another vector field G D .g1; : : : ; gN / such that(9.2) holds. Then, for fixed j 2 f1; : : : ; N g, we have

Z

I

f j dy D

Z bj

aj

Z

I�j

f j dHN�1 d�j

D

Z bj

aj

Z

I�j

gj dHN�1 d�j D

Z

I

gj dy

(9.26)

for any cube I . This implies that

f j .y/ D gj .y/ for almost every y:

�

Remark 9.7. In the proof of Theorem 9.4, working with QE WD E0 [ @�E and�S D .S; �/, we obtain (see Corollary 6.1) that

F.�S/ D �

Z

S

QFi � � D

Z

S

Fe � �;

where QFi �� and Fe �� are the interior and exterior normal traces of F relative to QE

and E, respectively. That is, the normal traces of F 2 DM1loc.�/ are the Cauchy

densities over all oriented surfaces.


10 Mathematical Formulation of the Balance Law

and Derivation of Systems of Balance Laws

In this section we first present the mathematical formulation for the physicalprinciple of balance law (9.1). Then we apply the results established in Sections 3through 9 to give a rigorous derivation of systems of balance laws with measuresource terms. In particular, we give a derivation of hyperbolic systems of conser-vation laws (10.11).

A balance law on an open subset � of RN postulates that the production of

a vector-valued “extensive” quantity in any bounded measurable subset E b �

with finite perimeter is balanced by the Cauchy flux of this quantity through themeasure-theoretic boundary @mE of E (see Dafermos [22, 23]).

Like the Cauchy flux, the production is introduced through a functional P , de-fined on any bounded measurable subset of finite perimeter E � �, taking valuein R

k and satisfying the conditions

P.E1 [E2/ D P.E1/C P.E2/ if E1 \E2 D ¿;(10.1)

jP.E/j � �.E/:(10.2)

Then the physical principle of balance law can be mathematically formulated as

(10.3) F.@mE/ D P.E/

for any bounded measurable subset of finite perimeter E � �.Fuglede’s theorem, Theorem 9.5, indicates that conditions (10.1) and (10.2)

imply that there is a production density P 2 M.�I Rk/ such that

(10.4) P.E/ D

Z

E1

P.y/:

On the other hand, combining Theorem 5.2 with the argument in Section 9yields that there exists F 2 DM1

loc.�I RN�k/ such that

(10.5) F.@mE/ D �

Z

@mE

.Fi � �/dHN�1 D

Z

E1

div F .y/

for any set of finite perimeter E b �.Then (10.3)–(10.5) yields the system of field equations

(10.6) div F .y/ D P.y/

in the sense of measures on �.

We assume that the state of the medium is described by a state vector field u,taking value in an open subset U of R

k , which determines both the flux densityfield F and the production density field P at the point y 2 � by the constitutive

equations

(10.7) F .y/ WD F .u.y/; y/; P.y/ WD P.u.y/; y/;


where F .u; y/ and P.u; y/ are given smooth functions defined on U ��.Combining (10.6) with (10.7) leads to the first-order, quasi-linear system of

partial differential equations

(10.8) div F .u.y/; y/ D P.u.y/; y/;

which is called a system of balance laws (cf. [22]).If P D 0, the previous derivation yields

(10.9) div F .u.y/; y/ D 0;

which is called a system of conservation laws. When the medium is homogeneous

F .u; y/ D F .u/I

that is, F depends on y only through the state vector. Then system (10.9) becomes

(10.10) div F .u.y// D 0:

In particular, when the coordinate system y is described by the time variable tand the space variable x D .x1; : : : ; xn/,

y D .t; x1; : : : ; xn/ D .t; x/; N D nC 1;

and the flux density is written as

F .u/ D .u; f1.u/; : : : ; fn.u// D .u; f .u//;

then we have the following standard form for the system of conservation laws:

(10.11) @tuC rx � f .u/ D 0; x 2 Rn; u 2 R

k :

11 Entropy Solutions of Hyperbolic Conservation Laws

We now apply the results established in Sections 3 through 9 to the recoveryof Cauchy entropy fluxes through the Lax entropy inequality for entropy solutionsof hyperbolic conservation laws by capturing entropy dissipation. We focus onsystem (10.11), which is assumed to be hyperbolic.

DEFINITION 11.1 A function � W Rk ! R is called an entropy of (10.11) if there

exists q W Rk ! R

n such that

(11.1) rqj .u/ D r�.u/rfj .u/; j D 1; : : : ; n:

Then the vector function q.u/ is called an entropy flux associated with the en-tropy �.u/, and the pair .�.u/; q.u// is called an entropy pair. The entropy pair.�.u/; q.u// is called a convex entropy pair on the domain U � R

k if the Hes-sian matrix r2�.u/ � 0 for any u 2 U . The entropy pair .�.u/; q.u// is called astrictly convex entropy pair on the domain U if r2�.u/ > 0 for any u 2 U .

Friedrichs and Lax [37] observed that most systems of conservation laws thatresult from continuum mechanics are endowed with a globally defined, strictlyconvex entropy. The available existence theories show that solutions of (10.11)generally fall within the following class of entropy solutions.


DEFINITION 11.2 A vector function u D u.t; x/ 2 L1loc.RC � R

n/ is called anentropy solution of (10.11) if u.t; x/ satisfies the Lax entropy inequality

(11.2) @t�.u.t; x//C rx � q.u.t; x// � 0

in the sense of distributions for any convex entropy pair .�; q/ W Rk ! R � R

n.

Clearly, an entropy solution is a weak solution by choosing �.u/ D ˙u in(11.2).

One of the main issues in conservation laws is to study the behavior of entropysolutions in this class to explore to the fullest extent possible all questions relatingto large-time behavior, uniqueness, stability, structure, and traces of entropy solu-tions, with neither specific reference to any particular method for constructing thesolutions nor additional regularity assumptions. Because the distribution

@t�.u.t; x//C rx � q.u.t; x//

is nonpositive, we conclude that it is in fact a Radon measure; that is, the field.�.u.t; x//; q.u.t; x/// is a divergence-measure field. Thus there exists �� 2

M.RC � Rn/ with �� 0 such that

(11.3) div.t;x/.�.u.t; x//; q.u.t; x/// D ��:

For any L1 entropy solution u, it was first indicated in Chen [14] that if thesystem is endowed with a strictly convex entropy, then, for any C 2 entropy pair.�; q/, there exists �� 2 M.RC � R

n/ such that

(11.4) div.t;x/.�.u.t; x//; q.u.t; x/// D ��:

We introduce a functional on any oriented surface S ,

(11.5) F�.S/ D

Z

S

.�.u/; q.u// � � dHn;

where .�.u/; q.u// � � is the normal trace in the sense of Theorem 5.2, since.�.u/; q.u// 2 DM1

loc.RC �Rn/. It is easy to check that the functional F� defined

by (11.5) is a Cauchy flux in the sense of Definition 9.3.

DEFINITION 11.3 (Cauchy Entropy Fluxes) A functional F� defined by (11.5) iscalled a Cauchy entropy flux with respect to the entropy �.

In particular, when � is convex,

F�.S/ � 0

for any oriented surface S . Furthermore, we can reformulate the balance law ofentropy from the recovery of an entropy production by capturing entropy dissipa-tion.


Moreover, it is clear that understanding more properties of divergence-measurefields can advance our understanding of the behavior of entropy solutions for hy-perbolic conservation laws and other related nonlinear equations by selecting ap-propriate entropy pairs. As examples, we refer the reader to [15, 16, 17, 19] forthe stability of Riemann solutions, which may contain rarefaction waves, contactdiscontinuities, and/or vacuum states, in the class of entropy solutions of the Eulerequations for gas dynamics; to [15, 18] for the decay of periodic entropy solu-tions for hyperbolic conservation laws; to [20, 65] for the initial and boundarylayer problems for hyperbolic conservation laws; to [16, 21] for the initial bound-ary value problems for hyperbolic conservation laws; and to [11, 52] for nonlineardegenerate parabolic-hyperbolic equations.

It is hoped that the theory of divergence-measure fields can be used to de-velop techniques in entropy methods, measure-theoretic analysis, partial differ-ential equations, and related areas.

Acknowledgment. The authors would like to thank Luis Caffarelli, Constan-tine Dafermos, Willi Jäger, Fanghua Lin, Leon Simon, and David Swanson forstimulating and fruitful discussions. Gui-Qiang Chen’s research was supported inpart by the National Science Foundation under grants DMS-0505473 andDMS-0244473 and an Alexander von Humboldt Foundation fellowship. MonicaTorres’s research was supported in part by the National Science Foundation undergrant DMS-0540869.

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GUI-QIANG CHEN

Department of MathematicsNorthwestern University2033 Sheridan RoadEvanston, IL 60208-2730E-mail: gqchen@math.

northwestern.edu

WILLIAM P. ZIEMER

Department of MathematicsIndiana UniversityRawles HallBloomington, IN 47405E-mail: [email protected]

MONICA TORRES

Department of MathematicsPurdue University150 N. University StreetWest Layayette, IN 47907-2067E-mail: [email protected].

edu

Received September 2007.Revised March 2008.

gauss-green theorem for weakly differentiable vector...

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